Difference between revisions of "2020 AMC 10A Problems/Problem 24"

Latest revision as of 18:33, 31 August 2024

Problem

Let $n$ be the least positive integer greater than $1000$ for which

$\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.$

What is the sum of the digits of $n$ ?

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$

Solution 1

We know that $\gcd(n+57,63)=21$ and $\gcd(n-57, 120)= 60$ by the Euclidean Algorithm. Hence, let $n+57=21\alpha$ and $n-57=60 \gamma$ , where $\gcd(\alpha,3)=1$ and $\gcd(\gamma,2)=1$ . Subtracting the two equations, $38=7\alpha-20\gamma$ . Letting $\gamma = 2s+1$ , we get $58=7\alpha-40s$ . Taking modulo $40$ , we have $\alpha \equiv{14} \pmod{40}$ . We are given that $n=21\alpha -57 >1000$ , so $\alpha \geq 51$ . Notice that if $\alpha =54$ then the condition $\gcd(\alpha,3)=1$ is violated. The next possible value of $\alpha = 94$ satisfies the given condition, giving us $n=1917$ .

Alternatively, we could have said $\alpha = 40k+14 \equiv{0} \pmod{3}$ for $k \equiv{1} \pmod{3}$ only, so $k \equiv{0,2} \pmod{3}$ , giving us our answer. Since the problem asks for the sum of the digits of $n$ , our answer is $1+9+1+7=\boxed{\textbf{(C) } 18}$ .

~Prabh1512

Solution 2

The conditions of the problem reduce to the following: $n+120 = 21k$ and $n+63 = 60\ell$ , where $\gcd(k,3) = 1$ and $\gcd(\ell,2) = 1$ . From these equations, we see that $21k - 60\ell = 57$ . Solving this Diophantine equation gives us that $k = 20a + 17$ and $\ell = 7a + 5$ . Since, $n>1000$ , we can do some bounding and get that $k > 53$ and $\ell > 17$ . Now we start bashing by plugging in numbers that satisfy these conditions: $a=4$ is the first number that works so we get $\ell = 33$ , $k = 97$ . Therefore, we have $n=21(97)-120=60(33)-63=1917$ , and our answer is $1+9+1+7=\boxed{\textbf{(C) } 18}$ .

Solution 3

We first find that $n\equiv6\pmod{21}$ and $n\equiv57\pmod{60}$ , then we get $n=21x+6$ and $n=60y+57$ by definitions, where $x$ and $y$ are integers. It follows that $y$ must be odd, since the GCD will be $120$ instead of $60$ if $y$ is even. Also, the unit digit of $n$ must be $7$ , since the unit digit of $60y$ is always $0$ and the unit digit of $57$ is $7$ . Therefore, we find that $x$ must end in $1$ to satisfy $n$ having a unit digit of $7$ . Also, we find that $x$ must not be a multiple of $3$ or else the GCD will be $63$ . Therefore, we test values for $x$ and find that $x=91$ satisfies all these conditions. Therefore, $n=1917$ and $1+9+1+7 = \boxed{\textbf{(C) } 18}$ .

~happykeeper

Solution 4

We are given that $\gcd(63, n+120) =21$ and $\gcd(n+63, 120)=60.$ By applying the Euclidean algorithm in reverse, we have $\gcd(63, n+120) = \gcd(63, n+120 + 63) = \gcd(63, n+183) = 21$ and $\gcd(n+63, 120) = \gcd(n+63 + 120, 120) = \gcd(n+183, 120) = 60.$

We now know that $n+183$ must be divisible by $21$ and $60,$ so it is divisible by $\text{lcm}(21, 60) = 420.$ Therefore, $n+183 = 420k$ for some integer $k.$ We know that $3 \nmid k,$ or else the first condition won't hold ( $\gcd$ will be $63$ ) and $2 \nmid k,$ or else the second condition won't hold ( $\gcd$ will be $120$ ). Since $k = 1$ gives us too small of an answer, then $k=5,$ from which $n = 1917.$ So, the answer is $1+9+1+7 = \boxed{\textbf{(C) } 18}.$

Solution 5

$\gcd(n+63,120)=60$ tells us $n+63\equiv60\pmod {120}$ . The smallest $n+63$ that satisfies the previous condition and $n>1000$ is $1140$ , so we start from there. If $n+63=1140$ , then $n+120=1197$ . Because $\gcd(n+120,63)=21$ , $n+120\equiv21\pmod {63}$ or $n+120\equiv42\pmod {63}$ . We see that $1197\equiv0\pmod {63}$ , which does not fulfill the requirement for $n+120$ , so we continue by keep on adding $120$ to $1197$ , in order to also fulfill the requirement for $n+63$ . Soon, we see that $n+120\pmod {63}$ decreases by $6$ every time we add $120$ , so we can quickly see that $n=1917$ because at that point $n+120\equiv21\pmod {63}$ . We add up all digits of $1917$ to get $\boxed{\textbf{(C) } 18}$ .

~SmileKat32

Solution 6

We are able to set up the following system of congruences: $\begin{align*} n &\equiv 6 \pmod {21}, \\ n &\equiv 57 \pmod {60}. \end{align*}$ Therefore, by definition, we are able to set-up the following system of equations: $\begin{align*} n &= 21a + 6, \\ n &= 60b + 57. \end{align*}$ Thus, we have $21a + 6 = 60b + 57,$ from which $7a + 2 = 20b + 19.$ We know $7a \equiv 0 \pmod {7},$ and since $7a = 20b + 17,$ therefore $20b + 17 \equiv 0 \pmod{7}.$ Simplifying this congruence further, we have $b\equiv 3 \pmod{7}.$ Thus, by definition, $b = 7x + 3.$ Substituting this back into our original equation, we get $n = 60(7x + 3) + 57 = 420x + 237.$ By definition, we are able to set up the following congruence: $n \equiv 237 \pmod{420}.$ Thus, $n = 1917$ , so our answer is simply $1+9+1+7=\boxed{\textbf{(C) } 18}$ .

Remark

Since $n \equiv -120 \pmod{21},$ we conclude that $n \equiv 6 \pmod{21}.$

Since $n \equiv -63 \pmod{60},$ we conclude that $n \equiv 57 \pmod{60}.$

Remember that $b \equiv 3 \pmod{7}.$

Lastly, the reason why $n \neq 1077$ is $n + 120$ would be divisible by $63$ , which is not possible due to the certain condition.

~nikenissan ~Midnight

Solution 7

First, we find $n$ . We know that it is greater than $1000$ , so we first input $n = 1000$ . From the first equation, $\gcd(63, n + 120) = 21$ , we know that if $n$ is correct, after we add $120$ to it, it should be divisible by $21$ , but not $63$ : $\frac{n + 120}{21} = \frac{1120}{21} = 53\text{ R }7.$ This does not work. To get to the nearest number divisible by $21$ , we have to add $14$ to cancel out the remainder. (Note that we don't subtract $7$ to get to $53$ ; $n$ is already at its lowest possible value!) Adding $14$ to $1000$ gives us $n = 1014$ . (Note: $n$ is currently divisible by 63, but that's fine since we'll be changing it in the next step.)

Now using the second equation, $\gcd(n + 63, 120) = 60$ , we know that if $n$ is correct, then $n+63$ is divisible by $60$ but not $120$ : $\frac{n + 63}{60} = \frac{1077}{60} = 17\text{ R }57.$ Again, this does not work. This requires some guessing and checking. We can add $21$ over and over again until $n$ is valid. This changes $n$ while also maintaining that $\frac{n + 120}{21}$ has no remainders. After adding $21$ once, we get $18 r 18$ . By pure luck, adding $21$ two more times gives us $19$ with no remainders. We now have $1077 + 21 + 21 + 21 = 1140$ . However, this number is divisible by $120$ . To get the next possible number, we add the LCM of $21$ and $60$ (once again, to maintain divisibility), which is $420$ . Unfortunately, $1140 + 420 = 1560$ is still divisible by $120$ . Adding $420$ again gives us $1980$ , which is valid. However, remember that this is equal to $n + 63$ , so subtracting $63$ from $1980$ gives us $1917$ , which is $n$ .

The sum of its digits are $1 + 9 + 1 + 7 = \boxed{\textbf{(C) } 18}$ .

~primegn

Solution 8 (Euclidean Algorithm)

By the Euclidean Algorithm, we have $\begin{alignat*}{8} \gcd(63,n+120)&=\hspace{1mm}&&\gcd(63,\phantom{ }\underbrace{n+120-63k_1}_{(n+120) \ \mathrm{mod} \ 63}\phantom{ })&&=21, \\ \gcd(n+63,120)&=&&\gcd(\phantom{ }\underbrace{n+63-120k_2}_{(n+63) \ \mathrm{mod} \ 120}\phantom{ },120)&&=60. \end{alignat*}$ Clearly, $n+120-63k_1$ must be either $21$ or $42,$ and $n+63-120k_2$ must be $60.$

More generally, let $t\in\{1,2\},$ so we get $\begin{align*} n+120-63k_1&=21t, &\hspace{55.5mm}(1) \\ n+63-120k_2&=60. &\hspace{55.5mm}(2) \end{align*}$ Subtracting $(2)$ from $(1)$ and then simplifying give $\begin{align*} 57-63k_1+120k_2&=21t-60 \\ 117-63k_1+120k_2&=21t \\ 39-21k_1+40k_2&=7t. \hspace{54mm}(\bigstar) \end{align*}$ Taking $(\bigstar)$ modulo $7$ produces $\begin{align*} 4+5k_2&\equiv0\pmod{7} \\ k_2&\equiv2\pmod{7}. \end{align*}$ Recall that $n>1000.$ From $(2),$ it follows that $1063-120k_2<n+63-120k_2=60,$ from which $k_2>8.$ Therefore, the possible values for $k_2$ are $9,16,23,\ldots.$

We need to check whether positive integers $k_1$ and $t$ (where $t\in\{1,2\}$ ) exist in $(1):$

If $k_2=9,$ then substituting into $(2)$ gives $n=1077.$ Next, substituting into $(1)$ produces $1197-63k_1=21t,$ or $57-3k_1=t.$
There are no solutions $(k_1,t).$

If $k_2=16,$ then substituting into $(2)$ gives $n=1917.$ Next, substituting into $(1)$ produces $2037-63k_1=21t,$ or $97-3k_1=t.$
The solution is $(k_1,t)=(32,1).$

Finally, the least such positive integer $n$ is $1917.$ The sum of its digits is $1+9+1+7=\boxed{\textbf{(C) } 18}.$

~MRENTHUSIASM

Solution 9 (Euclidean Algorithm)

Because we are finding value of $n$ for $n > 1000$ , let $n = 1000 + k$ .

Using the Euclidean Algorithm, $\begin{align*} \gcd(63, n+120) &= \gcd(63, 1000 + k + 120) \\ &= \gcd(63, k + 1120 - 63 \cdot 18) \\ &= \gcd(63, k-14) \\ &= 21, \\ \gcd(n+63, 120) &= \gcd(k + 1000 + 63, 120) \\ &= \gcd(k + 1000 + 63 - 120 \cdot 9, 120) \\ &= \gcd(k-17, 120) \\ &= 60. \end{align*}$ So, we have $\begin{align*} k &\equiv 14 \pmod{21}, \\ k &\equiv 17 \pmod{60}. \end{align*}$ Let $k = 21p + 14 = 60q + 17$ , $7p= 20q + 1$ (by the Division Algorithm), $7p = 21q - (q - 1)$ , $q - 1$ is a multiple of $7$ .

Let $q-1 = 7r$ , $q = 7r+1$ , $k = 60(7r+1) + 17 = 420r + 77$ .

$n = 1000 + k = 420r + 1077$ , $n = 1077, 1497, 1917$ .

By substituting $1077$ , $1497$ , $1917$ into $\gcd(63, n+120) =21$ and $\gcd(n+63, 120)=60$ , $1077$ and $1497$ aren't valid answers, only $1917$ is. Therefore, the answer is $1+9+1+7=\boxed{\textbf{(C) } 18}$ .

~isabelchen (Solution)

~Plainoldnumbertheory (Minor Edits)

~Puck_0 (Formatting)

Solution 10 (Diophantine Equations)

We know $63 = 3 \cdot 21$ , and $\gcd(63, n + 120) = 21 \Longrightarrow n + 120 = 21a$ , where $a$ is not a multiple of $3$ .

Also, $120 = 2 \cdot 60$ , and $gcd(n+63, 120) = 60 \Longrightarrow n + 63 = 60b$ , where $b$ is not a multiple of $2$ .

Let $n+63 = 60b = x$ , $n = x - 63$ , $n+120 = x+57 = 21a$ .

Now the problem becomes $\gcd(63, x+57) =21$ and $\gcd(x, 120)=60$ .

Meaning $x + 57$ has to be a multiple of $21$ but not $63$ , and $x$ is a multiple of $60$ but not $120$ .

Using trial and error, the least values are $x = 33\cdot60 = 1980$ and $n = x-63 = 1917$ .

Therefore, the answer is $1+9+1+7=\boxed{\textbf{(C) } 18}$ .

~isabelchen

Solution 11 (Chinese Remainder Theorem)

We have $\begin{align*} \gcd(63, n+120) &= 21 \Longrightarrow n + 120 \equiv 0 \pmod{21}, \text{ where } 9 \nmid n + 120, \\ \gcd(n+63, 120) &= 60 \Longrightarrow n + 63 \equiv 0 \pmod{60}, \text{ where } 8 \nmid n + 63. \end{align*}$ So, we conclude that $n \equiv 6 \pmod{21}$ and $n \equiv 57 \pmod{60}$ , respectively.

Because the $2$ moduli $21$ and $60$ are not relatively prime, namely $\gcd{(21, 60)} = 3$ , $21 = 3 \cdot 7$ , and $60 = 3 \cdot 20$ , we convert the system of $2$ linear congruences with non-coprime moduli into a system of $3$ linear congruences with coprime moduli: $\begin{align*} n &\equiv 0 \pmod{3}, \\ n &\equiv 6 \pmod{7}, \\ n &\equiv 17 \pmod{20}. \end{align*}$ By Chinese Remainder Theorem, the general solution of system of $3$ linear congruences is $n = 420k + 1077.$ We construct the following table: $\begin{array}{c|c|c|c} n & 1077 & 1497 & 1917\\ \hline n + 120 & 1197 & 1617 & 2037\\ \hline n + 63 & 1140 & 1560 & 1980\\ \end{array}$ Only $n = 1917$ satisfies $9 \nmid n + 120$ and $8 \nmid n + 63$ . Therefore, the answer is $1+9+1+7=\boxed{\textbf{(C) } 18}$ .

~isabelchen

Solution 12 (Logically Bashing)

We know that $n$ ends with the number $7$ , since $n+63$ is divisible by $60$ , and any number divisible by $10$ (a factor of $60$ ) must end with a $0$ . As a result, $n+120$ must also end in $7$ . Because $n+63$ must be divisible by $20$ , as well (using the same gcd), the tens value of $n$ must be odd. Also, $n+120$ cannot be divisible by $9$ , otherwise, its gcd with $63$ will be $63$ , instead of $21$ .

Now, we can start bashing, using the $\gcd(63, n+120) =21$ . We are given that $n$ must be greater than $1000$ , so we can try $n+120$ that is greater than $1120$ .

We try $21 \cdot 67$ , but it has an even tens digit. We then can try the next highest, using $77$ , but the $\gcd(n+63, 120) =120$ , instead of $60$ .

Since $87$ is divisible by $3$ , so when multiplied by $21$ , it will be divisible by $9$ , we do not have to test it.

The next biggest is $97$ , which works and gives us $n=1917$ , which, when the digits are added up, equals $\boxed{\textbf{(C) } 18}$ .

~parmani

Video Solutions

Education The Study of Everything

Video Solution 4

https://www.youtube.com/watch?v=gdGmSyzR908&list=PLLCzevlMcsWNcTZEaxHe8VaccrhubDOlQ&index=5 ~ MathEx

Video Solution 5

https://youtu.be/R220vbM_my8?t=899 ~ amritvignesh0719062.0

Video Solution 6

https://youtu.be/YcXsdSKVywk

~savannahsolver

@@ Line 10: / Line 10: @@
 == Solution 1 ==
-We know that <math>(n+57,63)=21, (n-57, 120)= 60</math> by the Euclidean Algorithm. Hence, let <math>n+57=21\alpha, n-57=60 \gamma, (\alpha,3)=1, (\gamma,2)=1</math>. Subtracting the <math>2</math> equations, <math>38=7\alpha-20\gamma</math>. Letting <math>\gamma = 2s+1</math>, <math>58=7\alpha-40s</math>. Taking <math>\mod{40}</math>, we have <math>\alpha \equiv{14} \pmod{40}</math>. We are given <math>n=21\alpha -57 >1000 \implies \alpha \geq 51</math>. Notice that if <math>\alpha =54</math> then the condition <math>(\alpha,3)=1</math> is violated. The next possible value of  <math>\alpha = 94</math> satisfies the given condition, giving us the answer <math>\boxed{1917}</math>. Alternatively, we could have said <math>\alpha = 40k+14 \equiv{0} \pmod{3}</math> for <math>k \equiv{1} \pmod{3}</math> only, so <math>k \equiv{0,2} \pmod{3}</math>, giving us our answer. Since the problem asks for the sum of the digits of <math>n</math>, <math>1+9+1+7=\boxed{\textbf{(C) } 18}</math> is our answer.
+We know that <math>\gcd(n+57,63)=21</math> and <math>\gcd(n-57, 120)= 60</math> by the Euclidean Algorithm. Hence, let <math>n+57=21\alpha</math> and <math>n-57=60 \gamma</math>, where <math>\gcd(\alpha,3)=1</math> and <math>\gcd(\gamma,2)=1</math>. Subtracting the two equations, <math>38=7\alpha-20\gamma</math>. Letting <math>\gamma = 2s+1</math>, we get <math>58=7\alpha-40s</math>. Taking modulo <math>40</math>, we have <math>\alpha \equiv{14} \pmod{40}</math>. We are given that <math>n=21\alpha -57 >1000</math>, so <math>\alpha \geq 51</math>. Notice that if <math>\alpha =54</math> then the condition <math>\gcd(\alpha,3)=1</math> is violated. The next possible value of  <math>\alpha = 94</math> satisfies the given condition, giving us <math>n=1917</math>.
-~Prabh1512, with edits by Terribleteeth.
+Alternatively, we could have said <math>\alpha = 40k+14 \equiv{0} \pmod{3}</math> for <math>k \equiv{1} \pmod{3}</math> only, so <math>k \equiv{0,2} \pmod{3}</math>, giving us our answer. Since the problem asks for the sum of the digits of <math>n</math>, our answer is <math>1+9+1+7=\boxed{\textbf{(C) } 18}</math>.
-==Solution 2 (Bashing)==
+~Prabh1512
-We are given that <math>\gcd(63, n+120)=21</math> and <math>\gcd(n+63,120) = 60</math>. This tells us that <math>n+120</math> is divisible by <math>21</math> but not <math>63</math>. It also tells us that <math>n+63</math> is divisible by 60 but not 120. Starting, we find the least value of <math>n+120</math> which is divisible by <math>21</math> which satisfies the conditions for <math>n</math>, which is <math>1134</math>, making <math>n=1014</math>. We then now keep on adding <math>21</math> until we get a number which satisfies the second equation. This number turns out to be <math>1917</math>, whose digits add up to <math>\boxed{\textbf{(C) } 18}</math>.
-~Midnight
+==Solution 2==
+The conditions of the problem reduce to the following: <math>n+120 = 21k</math> and <math>n+63 = 60\ell</math>, where <math>\gcd(k,3) = 1</math> and <math>\gcd(\ell,2) = 1</math>. From these equations, we see that <math>21k - 60\ell = 57</math>. Solving this Diophantine equation gives us that <math>k = 20a + 17</math> and <math>\ell = 7a + 5</math>. Since, <math>n>1000</math>, we can do some bounding and get that <math>k > 53</math> and <math>\ell > 17</math>. Now we start bashing by plugging in numbers that satisfy these conditions: <math>a=4</math> is the first number that works so we get <math>\ell = 33</math>, <math>k = 97</math>. Therefore, we have <math>n=21(97)-120=60(33)-63=1917</math>, and our answer is <math>1+9+1+7=\boxed{\textbf{(C) } 18}</math>.
 ==Solution 3==
-The conditions of the problem reduce to the following. <math>n+120 = 21k</math> where <math>gcd(k,3) = 1</math> and <math>n+63 = 60l</math> where <math>gcd(l,2) = 1</math>. From these equations, we see that <math>21k - 60l = 57</math>. Solving this Diophantine equation gives us that <math>k = 20a + 17</math>, <math>l = 7a + 5</math> form. Since, <math>n</math> is greater than <math>1000</math>, we can do some bounding and get that <math>k > 53</math> and <math>l > 17</math>. Now we start the bash by plugging in numbers that satisfy these conditions. <math>a=4</math> is the first number that works so we get <math>l = 33</math>, <math>k = 97</math>. <math>n=21(97)-120=60(33)-63=1917</math>. Our answer is then <math>1+9+1+7=\boxed{\textbf{(C) } 18}</math>.
+We first find that <math>n\equiv6\pmod{21}</math> and <math>n\equiv57\pmod{60}</math>, then we get <math>n=21x+6</math> and <math>n=60y+57</math> by definitions, where <math>x</math> and <math>y</math> are integers. It follows that <math>y</math> must be odd, since the GCD will be <math>120</math> instead of <math>60</math> if <math>y</math> is even. Also, the unit digit of <math>n</math> must be <math>7</math>, since the unit digit of <math>60y</math> is always <math>0</math> and the unit digit of <math>57</math> is <math>7</math>. Therefore, we find that <math>x</math> must end in <math>1</math> to satisfy <math>n</math> having a unit digit of <math>7</math>. Also, we find that <math>x</math> must not be a multiple of <math>3</math> or else the GCD will be <math>63</math>. Therefore, we test values for <math>x</math> and find that <math>x=91</math> satisfies all these conditions. Therefore, <math>n=1917</math> and <math>1+9+1+7 = \boxed{\textbf{(C) } 18}</math>.
+~happykeeper
 ==Solution 4==
-You can first find that n must be congruent to <math>6\equiv0\pmod {21}</math> and <math>57\equiv0\pmod {60}</math>. The we can find that <math>n=21x+6</math> and <math>n=60y+57</math>, where x and y are integers. Then we can find that y must be odd, since if it was even the gcd will be 120, not 60. Also, the unit digit of n has to be 7, since the unit digit of 60y is always 0 and the unit digit of 57 is 7. Therefore, you can find that x must end in 1 to satisfy n having a unit digit of 7. Also, you can find that x must not be a multiple of three or else the gcd will be 63. Therefore, you can test values for x and you can find that x=91 satisfies all these conditions.Therefore, n is 1917 and <math>1+9+1+7 = \boxed{\textbf{(C) } 18}</math>.-happykeeper
+We are given that <math>\gcd(63, n+120) =21</math> and <math>\gcd(n+63, 120)=60.</math> By applying the Euclidean algorithm in reverse, we have <cmath>\gcd(63, n+120) = \gcd(63, n+120 + 63) = \gcd(63, n+183) = 21</cmath> and <cmath>\gcd(n+63, 120) = \gcd(n+63 + 120, 120) = \gcd(n+183, 120) = 60.</cmath>
-==Solution 5 (Reverse Euclidean Algorithm)==
-We are given that <math>\gcd(63, n+120) =21</math> and <math>\gcd(n+63, 120)=60.</math> By applying the Euclidean algorithm, but in reverse, we have <cmath>\gcd(63, n+120) = \gcd(63, n+120 + 63) = \gcd(63, n+183) = 21</cmath> and <cmath>\gcd(n+63, 120) = \gcd(n+63 + 120, 120) = \gcd(n+183, 120) = 60.</cmath>
-We now know that <math>n+183</math> must be divisible by <math>21</math> and <math>60,</math> so it is divisible by <math>\text{lcm}(21, 60) = 420.</math> Therefore, <math>n+183 = 420k</math> for some integer <math>k.</math> We know that <math>3 \nmid k,</math> or else the first condition won't hold (<math>\gcd</math> will be <math>63</math>) and <math>2 \nmid k,</math> or else the second condition won't hold (<math>\gcd</math> will be <math>120</math>). Since <math>k = 1</math> gives us too small of an answer, then <math>k=5 \implies n = 1917,</math> so the answer is <math>1+9+1+7 = \boxed{\textbf{(C) } 18}.</math>
+We now know that <math>n+183</math> must be divisible by <math>21</math> and <math>60,</math> so it is divisible by <math>\text{lcm}(21, 60) = 420.</math> Therefore, <math>n+183 = 420k</math> for some integer <math>k.</math> We know that <math>3 \nmid k,</math> or else the first condition won't hold (<math>\gcd</math> will be <math>63</math>) and <math>2 \nmid k,</math> or else the second condition won't hold (<math>\gcd</math> will be <math>120</math>). Since <math>k = 1</math> gives us too small of an answer, then <math>k=5,</math> from which <math>n = 1917.</math> So, the answer is <math>1+9+1+7 = \boxed{\textbf{(C) } 18}.</math>
-==Solution 6==
+==Solution 5==
-<math>\gcd(n+63,120)=60</math> tells us <math>n+63\equiv60\pmod {120}</math>. The smallest <math>n+63</math> that satisfies the previous condition and <math>n>1000</math> is <math>1140</math>, so we start from there. If <math>n+63=1140</math>, then <math>n+120=1197</math>. Because <math>\gcd(n+120,63)=21</math>, <math>n+120\equiv21\pmod {63}</math> or <math>n+120\equiv42\pmod {63}</math>. We see that <math>1197\equiv0\pmod {63}</math>, which does not fulfill the requirement for <math>n+120</math>, so we continue by keep on adding <math>120</math> to <math>1197</math>, in order to also fulfill the requirement for <math>n+63</math>. Soon, we see that <math>n+120\pmod {63}</math> decreases by <math>6</math> every time we add <math>120</math>, so we can quickly see that <math>n=1917</math> because at that point <math>n+120\equiv21\pmod {63}</math>. We add up all digits of <math>1917</math>: <math>\boxed{\textbf{(C) } 18}</math>.
+<math>\gcd(n+63,120)=60</math> tells us <math>n+63\equiv60\pmod {120}</math>. The smallest <math>n+63</math> that satisfies the previous condition and <math>n>1000</math> is <math>1140</math>, so we start from there. If <math>n+63=1140</math>, then <math>n+120=1197</math>. Because <math>\gcd(n+120,63)=21</math>, <math>n+120\equiv21\pmod {63}</math> or <math>n+120\equiv42\pmod {63}</math>. We see that <math>1197\equiv0\pmod {63}</math>, which does not fulfill the requirement for <math>n+120</math>, so we continue by keep on adding <math>120</math> to <math>1197</math>, in order to also fulfill the requirement for <math>n+63</math>. Soon, we see that <math>n+120\pmod {63}</math> decreases by <math>6</math> every time we add <math>120</math>, so we can quickly see that <math>n=1917</math> because at that point <math>n+120\equiv21\pmod {63}</math>. We add up all digits of <math>1917</math> to get <math>\boxed{\textbf{(C) } 18}</math>.
 ~SmileKat32
-==Solution 7==
+==Solution 6==
 We are able to set up the following system of congruences:
 <cmath>\begin{align*}
@@ Line 63: / Line 62: @@
 Lastly, the reason why <math>n \neq 1077</math> is <math>n + 120</math> would be divisible by <math>63</math>, which is not possible due to the certain condition.
-~nikenissan
+~nikenissan ~Midnight
-== Solution 8==
+== Solution 7==
 First, we find <math>n</math>. We know that it is greater than <math>1000</math>, so we first input <math>n = 1000</math>. From the first equation, <math>\gcd(63, n + 120) = 21</math>, we know that if <math>n</math> is correct, after we add <math>120</math> to it, it should be divisible by <math>21</math>, but not <math>63</math>:
 <cmath>\frac{n + 120}{21} = \frac{1120}{21} = 53\text{ R }7.</cmath>
-Uh oh. To get to the nearest number divisible by <math>21</math>, we have to add <math>14</math> to cancel out the remainder. (Note that we don't subtract <math>7</math> to get to <math>53</math>; <math>n</math> is already at its lowest possible value!)
+This does not work. To get to the nearest number divisible by <math>21</math>, we have to add <math>14</math> to cancel out the remainder. (Note that we don't subtract <math>7</math> to get to <math>53</math>; <math>n</math> is already at its lowest possible value!)
 Adding <math>14</math> to <math>1000</math> gives us <math>n = 1014</math>. (Note: <math>n</math> is currently divisible by 63, but that's fine since we'll be changing it in the next step.)
-Now using, the second equation, <math>\gcd(n + 63, 120) = 60</math>, we know that if <math>n</math> is correct, after we add <math>63</math> to it, it should be divisible by <math>60</math>, but not <math>120</math>:
+Now using the second equation, <math>\gcd(n + 63, 120) = 60</math>, we know that if <math>n</math> is correct, then <math>n+63</math> is divisible by <math>60</math> but not <math>120</math>:
 <cmath>\frac{n + 63}{60} = \frac{1077}{60} = 17\text{ R }57.</cmath>
-Uh oh (again). This requires some guessing and checking. We can add <math>21</math> over and over again until <math>n</math> is valid. This changes <math>n</math> while also maintaining that <math>\frac{n + 120}{21}</math> has no remainders.
+Again, this does not work. This requires some guessing and checking. We can add <math>21</math> over and over again until <math>n</math> is valid. This changes <math>n</math> while also maintaining that <math>\frac{n + 120}{21}</math> has no remainders.
 After adding <math>21</math> once, we get <math>18 r 18</math>. By pure luck, adding <math>21</math> two more times gives us <math>19</math> with no remainders.
 We now have <math>1077 + 21 + 21 + 21 = 1140</math>. However, this number is divisible by <math>120</math>. To get the next possible number, we add the LCM of <math>21</math> and <math>60</math> (once again, to maintain divisibility), which is <math>420</math>. Unfortunately, <math>1140 + 420 = 1560</math> is still divisible by <math>120</math>. Adding <math>420</math> again gives us <math>1980</math>, which is valid. However, remember that this is equal to <math>n + 63</math>, so subtracting <math>63</math> from <math>1980</math> gives us <math>1917</math>, which is <math>n</math>.
-The sum of its digits are <math>1 + 9 + 1 + 7 = \boxed{\textbf{(C) }18}</math>.
+The sum of its digits are <math>1 + 9 + 1 + 7 = \boxed{\textbf{(C) } 18}</math>.
 ~primegn
-==Solution 11 (Euclidean Algorithm)==
+==Solution 8 (Euclidean Algorithm)==
 By the Euclidean Algorithm, we have
 <cmath>\begin{alignat*}{8}
-\gcd(63,n+120)&=\hspace{1mm}&&\gcd(63,\phantom{ }\underbrace{n+120-63k_1}_{(n+120) \ \mathrm{mod} \ 63}\phantom{ })&&=21, &&\hspace{10mm}(1) \\
+\gcd(63,n+120)&=\hspace{1mm}&&\gcd(63,\phantom{ }\underbrace{n+120-63k_1}_{(n+120) \ \mathrm{mod} \ 63}\phantom{ })&&=21, \\
-\gcd(n+63,120)&=&&\gcd(\phantom{ }\underbrace{n+63-120k_2}_{(n+63) \ \mathrm{mod} \ 120}\phantom{ },120)&&=60.&&\hspace{10mm}(2)
+\gcd(n+63,120)&=&&\gcd(\phantom{ }\underbrace{n+63-120k_2}_{(n+63) \ \mathrm{mod} \ 120}\phantom{ },120)&&=60.
 \end{alignat*}</cmath>
 Clearly, <math>n+120-63k_1</math> must be either <math>21</math> or <math>42,</math> and <math>n+63-120k_2</math> must be <math>60.</math>
@@ Line 92: / Line 91: @@
 More generally, let <math>t\in\{1,2\},</math> so we get
 <cmath>\begin{align*}
-n+120-63k_1&=21t, &\hspace{55.5mm}(1*) \\
+n+120-63k_1&=21t, &\hspace{55.5mm}(1) \\
-n+63-120k_2&=60. &\hspace{55.5mm}(2*)
+n+63-120k_2&=60. &\hspace{55.5mm}(2)
 \end{align*}</cmath>
-Subtracting <math>(2*)</math> from <math>(1*)</math> and then simplifying give
+Subtracting <math>(2)</math> from <math>(1)</math> and then simplifying give
 <cmath>\begin{align*}
 -63k_1+120k_2&=21t-60 \\
@@ Line 106: / Line 105: @@
 k_2&\equiv2\pmod{7}.
 \end{align*}</cmath>
-Recall that <math>n>1000.</math> From <math>(2*),</math> it follows that <cmath>1063-120k_2<n+63-120k_2=60,</cmath> from which <math>k_2>8.</math> Therefore, the possible values for <math>k_2</math> are <math>9,16,23,\ldots.</math>
+Recall that <math>n>1000.</math> From <math>(2),</math> it follows that <cmath>1063-120k_2<n+63-120k_2=60,</cmath> from which <math>k_2>8.</math> Therefore, the possible values for <math>k_2</math> are <math>9,16,23,\ldots.</math>
-We need to check whether positive integers <math>k_1</math> and <math>t</math> (where <math>t\in\{1,2\}</math>) exist in <math>(1*):</math>
+We need to check whether positive integers <math>k_1</math> and <math>t</math> (where <math>t\in\{1,2\}</math>) exist in <math>(1):</math>
 <ul style="margin-left: 1.5em, list-style-type: square;">
-   <li>If <math>k_2=9,</math> then substituting into <math>(2*)</math> gives <math>n=1077.</math> Next, substituting into <math>(1*)</math> produces <math>1197-63k_1=21t,</math> or <math>57-3k_1=t.</math> <p>
+   <li>If <math>k_2=9,</math> then substituting into <math>(2)</math> gives <math>n=1077.</math> Next, substituting into <math>(1)</math> produces <math>1197-63k_1=21t,</math> or <math>57-3k_1=t.</math> <p>
 There are no solutions <math>(k_1,t).</math></li><p>
-   <li>If <math>k_2=16,</math> then substituting into <math>(2*)</math> gives <math>n=1917.</math> Next, substituting into <math>(1*)</math> produces <math>2037-63k_1=21t,</math> or <math>97-3k_1=t.</math> <p>
+   <li>If <math>k_2=16,</math> then substituting into <math>(2)</math> gives <math>n=1917.</math> Next, substituting into <math>(1)</math> produces <math>2037-63k_1=21t,</math> or <math>97-3k_1=t.</math> <p>
 The solution is <math>(k_1,t)=(32,1).</math></li><p>
 </ul>
@@ Line 119: / Line 118: @@
 ~MRENTHUSIASM
-== Solution 12 (Euclidean Algorithm) ==
+== Solution 9 (Euclidean Algorithm) ==
 Because we are finding value of <math>n</math> for <math>n > 1000</math>, let <math>n = 1000 + k</math>.
-Using the Euclidian Algorithm,
+Using the Euclidean Algorithm,
 <cmath>\begin{align*}
-\gcd(63, n+120) &= \gcd(63, 1000 + k + 120) = \gcd(63, k + 1120 - 63 \cdot 18) = \gcd(63, k-14) = 21, \\
+\gcd(63, n+120) &= \gcd(63, 1000 + k + 120) \\
-\gcd(n+63, 120) &= \gcd(k + 1000 + 63, 120) = \gcd(k + 1000 + 63 - 120 \cdot 9, 120) = \gcd(k-17, 120) = 60.
+&= \gcd(63, k + 1120 - 63 \cdot 18) \\
+&= \gcd(63, k-14) \\
+&= 21, \\
+\gcd(n+63, 120) &= \gcd(k + 1000 + 63, 120) \\
+&= \gcd(k + 1000 + 63 - 120 \cdot 9, 120) \\
+&= \gcd(k-17, 120) \\
+&= 60.
 \end{align*}</cmath>
 So, we have
@@ Line 133: / Line 138: @@
 k &\equiv 17 \pmod{60}.
 \end{align*}</cmath>
-Let <math>k = 21p + 14 = 60q + 17</math>, <math>7p= 20q + 1</math>, <math>7p = 21q - (q - 1)</math>, <math>q - 1</math> is a multiple of <math>7</math>.
+Let <math>k = 21p + 14 = 60q + 17</math>, <math>7p= 20q + 1</math> (by the Division Algorithm), <math>7p = 21q - (q - 1)</math>, <math>q - 1</math> is a multiple of <math>7</math>.
 Let <math>q-1 = 7r</math>, <math>q = 7r+1</math>, <math>k = 60(7r+1) + 17 = 420r + 77</math>.
@@ Line 141: / Line 146: @@
 By substituting <math>1077</math>, <math>1497</math>, <math>1917</math> into <math>\gcd(63, n+120) =21</math> and <math>\gcd(n+63, 120)=60</math>, <math>1077</math> and  <math>1497</math> aren't valid answers, only <math>1917</math> is. Therefore, the answer is <math>1+9+1+7=\boxed{\textbf{(C) } 18}</math>.
-~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] (Solution)
+~[https://artofproblemsolving.com/wiki/index.php/User:Plainoldnumbertheory Plainoldnumbertheory] (Minor Edits)
+~[https://artofproblemsolving.com/wiki/index.php/User:Plainoldnumbertheory Puck_0] (Formatting)
-== Solution 13 (Diophantine Equations)==
+== Solution 10 (Diophantine Equations)==
 We know <math>63 = 3 \cdot 21</math>, and <math>\gcd(63, n + 120) = 21 \Longrightarrow n + 120 = 21a</math>, where <math>a</math> is not a multiple of <math>3</math>.
@@ Line 161: / Line 170: @@
 ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
-== Solution 14 (Chinese Remainder Theorem) ==
+== Solution 11 (Chinese Remainder Theorem) ==
 We have
 <cmath>\begin{align*}
@@ Line 187: / Line 196: @@
 ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+== Solution 12 (Logically Bashing) ==
+We know that <math>n</math> ends with the number <math>7</math>, since <math>n+63</math> is divisible by <math>60</math>, and any number divisible by <math>10</math> (a factor of <math>60</math>) must end with a <math>0</math>. As a result, <math>n+120</math> must also end in <math>7</math>. Because <math>n+63</math> must be divisible by <math>20</math>, as well (using the same gcd), the tens value of <math>n</math> must be odd. Also, <math>n+120</math> cannot be divisible by <math>9</math>, otherwise, its gcd with <math>63</math> will be <math>63</math>, instead of <math>21</math>.
+Now, we can start bashing, using the <math>\gcd(63, n+120) =21</math>. We are given that <math>n</math> must be greater than <math>1000</math>, so we can try <math>n+120</math> that is greater than <math>1120</math>.
+We try <math>21 \cdot 67</math>, but it has an even tens digit. We then can try the next highest, using <math>77</math>, but the <math>\gcd(n+63, 120) =120</math>, instead of <math>60</math>.
+Since <math>87</math> is divisible by <math>3</math>, so when multiplied by <math>21</math>, it will be divisible by <math>9</math>, we do not have to test it.
+The next biggest is <math>97</math>, which works and gives us <math>n=1917</math>, which, when the digits are added up, equals <math>\boxed{\textbf{(C) } 18}</math>.
+~parmani
 ==Video Solutions==
 ===Video Solution 1 (Richard Rusczyk)===
-https://artofproblemsolving.com/videos/amc/2020amc10a/514
+https://www.youtube.com/watch?v=tk3yOGG2K-s&
 ===Video Solution 2===
@@ Line 205: / Line 227: @@
 ===Video Solution 5===
 https://youtu.be/R220vbM_my8?t=899 ~ amritvignesh0719062.0
+==Video Solution 6==
+https://youtu.be/YcXsdSKVywk
+~savannahsolver
 ==See Also==

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

Page

Toolbox

Search