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

## 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 $(n+57,63)=21, (n-57, 120)= 60$ by the Euclidean Algorithm. Hence, let $n+57=21\alpha, n-57=60 \gamma, (\alpha,3)=1, (\gamma,2)=1$. Subtracting the $2$ equations, $38=7\alpha-20\gamma$. Letting $\gamma = 2s+1$, $58=7\alpha-40s$. Taking $\mod{40}$, we have $\alpha \equiv{14} \pmod{40}$. We are given $n=21\alpha -57 >1000 \implies \alpha \geq 51$. Notice that if $\alpha =54$ then the condition $(\alpha,3)=1$ is violated. The next possible value of $\alpha = 94$ satisfies the given condition, giving us the answer $\boxed{1917}$. 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 $n$, $1+9+1+7$ = $18$ or $\boxed{\textbf{(C) } 18}$ is our answer.

~Prabh1512, with edits by Terribleteeth.

## Solution 2

We know that $\text{gcd}(63, n+120)=21$, so we can write $n+120\equiv0\pmod {21}$. Simplifying, we get $n\equiv6\pmod {21}$. Similarly, we can write $n+63\equiv0\pmod {60}$, or $n\equiv-3\pmod {60}$. Solving these two modular congruences, $n\equiv237\pmod {420}$ which we know is the only solution by the Chinese Remainder Theorem. Now, since the problem is asking for the least positive integer greater than $1000$, we find the least solution is $n=1077$.

However, we have not considered cases where $\text{gcd}(63, n+120)=63$ or $\text{gcd}(n+63, 120)=120$.

${1077+120} \equiv 0 \pmod {63}$ so we try $n=1077+420=1497$. ${1497+63}\equiv0\pmod {120}$ so again we add $420$ to $n$. It turns out that $n=1497+420=1917$ does indeed satisfy the original conditions, so our answer is $1+9+1+7=\boxed{\textbf{(C) }18}$.

## Solution 3 (Bashing)

We are given that $\gcd(63, n+120)=21$ and $\gcd(n+63,120) = 60$. This tells us that $n+120$ is divisible by $21$ but not $63$. It also tells us that $n+63$ is divisible by 60 but not 120. Starting, we find the least value of $n+120$ which is divisible by $21$ which satisfies the conditions for $n$, which is $1134$, making $n=1014$. We then now keep on adding $21$ until we get a number which satisfies the second equation. This number turns out to be $1917$, whose digits add up to $\boxed{\textbf{(C) } 18}$.

-Midnight

## Solution 4 (Bashing but Worse)

Assume that $n$ has 4 digits. Then $n = abcd$, where $a$, $b$, $c$, $d$ represent digits of the number (not to get confused with $a * b * c * d$). As given the problem, $gcd(63, n + 120) = 21$ and $gcd(n + 63, 120) = 60$. So we know that $d = 7$ (last digit of $n$). That means that $12 + abc \equiv0\pmod {7}$ and $7 + abc\equiv0\pmod {6}$. We can bash this after this. We just want to find all pairs of numbers $(x, y)$ such that $x$ is a multiple of 7 that is $5$ greater than a multiple of $6$. Our equation for $12 + abc$ would be $42*j + 35 = x$ and our equation for $7 + abc$ would be $42*j + 30 = y$, where $j$ is any integer. We plug this value in until we get a value of $abc$ that makes $n = abc7$ satisfy the original problem statement (remember, $abc > 100$). After bashing for hopefully a couple minutes, we find that $abc = 191$ works. So $n = 1917$ which means that the sum of its digits is $\boxed{\textbf{(C) } 18}$.

~ Baolan

## Solution 5

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

## Solution 6

You can first find that n must be congruent to $6\equiv0\pmod {21}$ and $57\equiv0\pmod {60}$. The we can find that $n=21x+6$ and $n=60y+57$, 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 $1+9+1+7 = \boxed{\textbf{(C) } 18}$.-happykeeper

## Solution 7 (Reverse Euclidean Algorithm)

We are given that $\gcd(63, n+120) =21$ and $\gcd(n+63, 120)=60.$ By applying the Euclidean algorithm, but 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 $k=5 \implies n = 1917,$ so the answer is $1+9+1+7 = \boxed{\textbf{(C) } 18}.$

## Solution 8

$\gcd(n+63,120)=60$ 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 $n+120\equiv21\pmod {63}$. Adding up all the digits in $1917$, we have $\boxed{\textbf{(C) } 18}$.

-SmileKat32

## Solution 9

We are able to set-up the following system-of-congruences: $$n \equiv 6 \pmod {21},$$ $$n \equiv 57 \pmod {60}.$$ Therefore, by definition, we are able to set-up the following system of equations: $$n = 21a + 6,$$ $$n = 60b + 57.$$ Thus, $$21a + 6 = 60b + 57$$ $$\implies 7a + 2 = 20b + 19.$$ 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 $$5b \equiv 1 \pmod{7}$$ $$\implies b \equiv 3 \pmod {7}.$$ Thus, by definition, $b = 7x + 3.$ Substituting this back into our original equation, $$n = 60(7x + 3) + 57$$ $$\implies n = 420x + 180 + 57$$ $$\implies n = 420x + 237.$$ By definition, we are able to set-up the following congruence: $$n \equiv 237 \pmod{420}.$$ Thus, $n = 1917$, so our answer is simply $\boxed{18}$.

(Remarks. $n \equiv 6 \pmod{21}$ since $n \equiv -120 \pmod{21},$ by definition & $n \equiv 57 \pmod{60}$ since $n \equiv -63 \pmod{60},$ by definition.

Remember, $5b \equiv 1 \pmod{7} \implies 5b \equiv 15 \pmod{7} \implies b \equiv 3 \pmod{7}.$

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

## Solution 10

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 $63$. $$\frac{n + 120}{21},$$ $$\frac{1120}{21},$$ $$53 r 7.$$ Uh oh. 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, after we add $63$ to it, it should be divisible by $60$, but not $120$. $$\frac{n + 63}{60},$$ $$\frac{1077}{60},$$ $$17r57.$$ Uh oh (again). 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 $1 + 9 + 1 + 7 = 18$.

So, our answer is $\boxed{\textbf{(C) }18}$. ~ primegn

## Solution 11 (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, &&\hspace{10mm}(1) \\ \gcd(n+63,120)&=&&\gcd(\phantom{ }\underbrace{n+63-120k_2}_{(n+63) \ \mathrm{mod} \ 120}\phantom{ },120)&&=60.&&\hspace{10mm}(2) \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 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

## Video Solutions

### Video Solution 2

https://youtu.be/8mNMKH0T9W0 - Happytwin

### Video Solution 3 (Quick & Simple)

Education The Study of Everything

### Video Solution 5

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