Difference between revisions of "2018 AMC 10A Problems/Problem 22"

Latest revision as of 03:51, 6 August 2023

Problem

Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$ , $\gcd(b, c)=36$ , $\gcd(c, d)=54$ , and $70<\gcd(d, a)<100$ . Which of the following must be a divisor of $a$ ?

$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$

Solution 1

The GCD information tells us that $24$ divides $a$ , both $24$ and $36$ divide $b$ , both $36$ and $54$ divide $c$ , and $54$ divides $d$ . Note that we have the prime factorizations: $\begin{align*} 24 &= 2^3\cdot 3,\\ 36 &= 2^2\cdot 3^2,\\ 54 &= 2\cdot 3^3. \end{align*}$

Hence we have $\begin{align*} a &= 2^3\cdot 3\cdot w\\ b &= 2^3\cdot 3^2\cdot x\\ c &= 2^2\cdot 3^3\cdot y\\ d &= 2\cdot 3^3\cdot z \end{align*}$ for some positive integers $w,x,y,z$ . Now if $3$ divides $w$ , then $\gcd(a,b)$ would be at least $2^3\cdot 3^2$ which is too large, hence $3$ does not divide $w$ . Similarly, if $2$ divides $z$ , then $\gcd(c,d)$ would be at least $2^2\cdot 3^3$ which is too large, so $2$ does not divide $z$ . Therefore, $\gcd(a,d)=2\cdot 3\cdot \gcd(w,z)$ where neither $2$ nor $3$ divide $\gcd(w,z)$ . In other words, $\gcd(w,z)$ is divisible only by primes that are at least $5$ . The only possible value of $\gcd(a,d)$ between $70$ and $100$ and which fits this criterion is $78=2\cdot3\cdot13$ , so the answer is $\boxed{\textbf{(D) }13}$ .

Solution 2

We can say that $a$ and $b$ 'have' $2^3 \cdot 3$ , that $b$ and $c$ have $2^2 \cdot 3^2$ , and that $c$ and $d$ have $3^3 \cdot 2$ . Combining $1$ and $2$ yields $b$ has (at a minimum) $2^3 \cdot 3^2$ , and thus $a$ has $2^3 \cdot 3$ (and no more powers of $3$ because otherwise $\gcd(a,b)$ would be different). In addition, $c$ has $3^3 \cdot 2^2$ , and thus $d$ has $3^3 \cdot 2$ (similar to $a$ , we see that $d$ cannot have any other powers of $2$ ). We now assume the simplest scenario, where $a = 2^3 \cdot 3$ and $d = 3^3 \cdot 2$ . According to this base case, we have $\gcd(a, d) = 2 \cdot 3 = 6$ . We want an extra factor between the two such that this number is between $70$ and $100$ , and this new factor cannot be divisible by $2$ or $3$ . Checking through, we see that $6 \cdot 13$ is the only one that works. Therefore the answer is $\boxed{\textbf{(D) } 13}$

Solution by JohnHankock

Solution 3 (Better notation)

First off, note that $24$ , $36$ , and $54$ are all of the form $2^x\times3^y$ . The prime factorizations are $2^3\times 3^1$ , $2^2\times 3^2$ and $2^1\times 3^3$ , respectively. Now, let $a_2$ and $a_3$ be the number of times $2$ and $3$ go into $a$ , respectively. Define $b_2$ , $b_3$ , $c_2$ , and $c_3$ similarly. Now, translate the $lcm$ s into the following: $1) \min(a_2,b_2)=3$ $2) \min(a_3,b_3)=1$ $3) \min(b_2,c_2)=2$ $4) \min(b_3,c_3)=2$ $5) \min(c_2,d_2)=1$ $6) \min(c_3,d_3)=3$

From $4)$ , we see that $b_3 \geq 2$ , thus from $2)$ , $a_3 = 1$ . Similarly, from $3)$ , $c_2 \geq 2$ , thus from $5)$ , $d_2 = 1$ .

Note also that $d_3 \geq 3$ and $a_2 \geq 3$ . Therefore $\min(a_2, d_2) = 1$ $\min(a_3, d_3) = 1$ Thus, $\gcd(a, d) = 2 \times 3 \times k$ for some $k$ having no factors of $2$ or $3$ .

Since $70 < \gcd(a, d) < 100$ , the only values for k are $12, 13, 14, 15, 16$ , but all have either factors of $2$ or $3$ , except $\boxed{\textbf{(D)} 13}$ . And we're done.

~Rowechen Zhong ~MannsNotHot

Solution 4 (Fastest)

Notice that $\gcd (a,b,c,d)=\gcd(\gcd(a,b),\gcd(b,c),\gcd(c,d))=\gcd(24,36,54)=6$ , so $\gcd(d,a)$ must be a multiple of $6$ . The only answer choice that gives a value between $70$ and $100$ when multiplied by $6$ is $\boxed{\textbf{(D) } 13}$ . - mathleticguyyy + einstein

In the case where there can be 2 possible answers, we can do casework on $\gcd(d,a)$ ~Williamgolly

Solution 5

Since $\gcd (a,b) = 24$ , $a = 24j$ and $b = 24k$ for some positive integers $j,k$ such that $j$ and $k$ are relatively prime.

Similarly , since $\gcd (b,c) = 36$ , we have $b = 24k$ and $c=36m$ with the same criteria. However, since $24$ is not a multiple of $36$ , we must contribute an extra $3$ to $b$ in order to make it a multiple of $36$ . So, $k$ is a multiple of three, and it is relatively prime to $m$ .

Finally, $\gcd (c,d) = 54$ , so using the same logic, $m$ is a multiple of $3$ and is relatively prime to $n$ where $d = 54n$ .

Since we can't really do anything with these messy expressions, we should try some sample cases of $a,b,c$ and $d$ . Specifically, we let $j = 5, 7, 11, 13$ or $17$ , and see which one works.

First we let $j= 5$ . Note that all of these values of $j$ work for the first $\gcd$ expression because they are all not divisible by $3$ .

Without the loss of generality, we let $k=m=3$ for all of our sample cases. We can also adjust the value of $n$ in $d=54n$ , since there is no fixed value for $\gcd(a,d)$ ; there is only a bound.

So we try to make our bound $70 < \gcd(a,d) < 100$ satisfactory. We do so by letting $j=n$ .

Testing our first case $a=24 \cdot 5$ and $d = 54 \cdot 5$ , we find that $\gcd(a,d) = 30$ . To simplify our work, we note that $\gcd(24,54) = 6$ , so $\gcd(24k, 54k)$ for all $k>1$ is equal to $6k$ .

So now, we can easily find our values of $\gcd(a,b)$ :

$\gcd(24 \cdot 5, 54 \cdot 5) = 6 \cdot 5 = 30$

$\gcd(24 \cdot 7, 54 \cdot 7) = 6 \cdot 7 = 42$

$\gcd(24 \cdot 11, 54 \cdot 11) = 6 \cdot 11 = 66$

$\boxed{\gcd(24 \cdot 13, 54 \cdot 13) = 6 \cdot 13 = 78}$

$\gcd(24 \cdot 17, 54 \cdot 17) = 6 \cdot 17 = 104$

We can clearly see that only $j=n=13$ is in the bound $70 < \gcd(a,d) < 100$ . So, $13$ must be a divisor of $a$ , which is answer choice $\boxed{\textbf{(D)}}$ .

-FIREDRAGONMATH16

Solution 6

$[asy] //Variable Declarations defaultpen(0.45); size(200pt); fontsize(15pt); pair X, Y, Z, W; real R; path quad; //Variable Definitions R = 1; X = R*dir(45); Y = R*dir(135); Z = R*dir(-135); W = R*dir(-45); quad = X--Y--Z--W--cycle; //Diagram draw(quad); label("$b$",X,NE); label("$a=2^3 \cdot 3 \cdot p$",Y,NW); label("$d=2 \cdot 3^3 \cdot q$",Z,SW); label("$c$",W,SE); label("$2^3 \cdot 3$",X--Y); label("$2^2 \cdot 3^2$",X--W); label("$2 \cdot 3^3$",Z--W); label("$2 \cdot 3 \cdot k$",Z--Y); [/asy]$

The relationship of $a$ , $b$ , $c$ , and $d$ is shown in the above diagram. $gcd(a,d)=2 \cdot 3 \cdot k$ . $70 < 6k < 100$ , $12 \le k \le 16$ , $k=\boxed{\textbf{(D) }13}$

Note that $gcd(b,c)=36$ is not required to solve the problem.

~isabelchen

Solution 7 (Easier version of Solution 1)

Just as in solution $1$ , we prime factorize $a, b, c$ and $d$ to observe that

$a=2^3\cdot{3}\cdot{w}$

$b=2^3\cdot{3^2}\cdot{x}$

$c=2^2\cdot{3^3}\cdot{y}$

$d=2\cdot{3^3}\cdot{z}.$

Substituting these expressions for $a$ and $d$ into the final given,

$70<\text{gcd}(2\cdot{3^3}\cdot{z}, 2^3\cdot{3}\cdot{w})<100.$

The greatest common divisor of these two numbers is already $6$ . If $k$ is what we wish to multiply $6$ by to obtain the gcd of these two numbers, then

$70<6k<100$ . Testing the answer choices, only $13$ works for $k$ (in order for the compound inequality to hold). so our gcd is $78$ , which means that $\boxed{\textbf{(D) }13}$ must divide $a$ .

-Benedict T (countmath1)

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2018amc10a/467

~ dolphin7

Video Solution

https://youtu.be/yjrqINsQP5c

~savannahsolver

Video Solution by OmegaLearn (Meta-Solving Technique)

https://youtu.be/GmUWIXXf_uk?t=1003

~ pi_is_3.14

@@ Line 27: / Line 27: @@
 == Solution 2 ==
-We can say that <math>a</math> and <math>b</math> 'have' <math>2^3 * 3</math>, that <math>b</math> and <math>c</math> have <math>2^2 * 3^2</math>, and that <math>c</math> and <math>d</math> have <math>3^3 * 2</math>. Combining <math>1</math> and <math>2</math> yields <math>b</math> has (at a minimum) <math>2^3 * 3^2</math>, and thus <math>a</math> has <math>2^3 * 3</math> (and no more powers of <math>3</math> because otherwise <math>gcd(a,b)</math> would be different). In addition, <math>c</math> has <math>3^3 * 2^2</math>, and thus <math>d</math> has <math>3^3 * 2</math> (similar to <math>a</math>, we see that <math>d</math> cannot have any other powers of <math>2</math>). We now assume the simplest scenario, where <math>a = 2^3 * 3</math> and <math>d = 3^3 * 2</math>. According to this base case, we have <math>gcd(a, d) = 2 * 3 = 6</math>. We want an extra factor between the two such that this number is between <math>70</math> and <math>100</math>, and this new factor cannot be divisible by <math>2</math> or <math>3</math>. Checking through, we see that <math>6 * 13</math> is the only one that works. Therefore the answer is <math>\boxed{\textbf{(D) } 13}</math>
+We can say that <math>a</math> and <math>b</math> 'have' <math>2^3 \cdot 3</math>, that <math>b</math> and <math>c</math> have <math>2^2 \cdot 3^2</math>, and that <math>c</math> and <math>d</math> have <math>3^3 \cdot 2</math>. Combining <math>1</math> and <math>2</math> yields <math>b</math> has (at a minimum) <math>2^3 \cdot 3^2</math>, and thus <math>a</math> has <math>2^3 \cdot 3</math> (and no more powers of <math>3</math> because otherwise <math>\gcd(a,b)</math> would be different). In addition, <math>c</math> has <math>3^3 \cdot 2^2</math>, and thus <math>d</math> has <math>3^3 \cdot 2</math> (similar to <math>a</math>, we see that <math>d</math> cannot have any other powers of <math>2</math>). We now assume the simplest scenario, where <math>a = 2^3 \cdot 3</math> and <math>d = 3^3 \cdot 2</math>. According to this base case, we have <math>\gcd(a, d) = 2 \cdot 3 = 6</math>. We want an extra factor between the two such that this number is between <math>70</math> and <math>100</math>, and this new factor cannot be divisible by <math>2</math> or <math>3</math>. Checking through, we see that <math>6 \cdot 13</math> is the only one that works. Therefore the answer is <math>\boxed{\textbf{(D) } 13}</math>
 Solution by JohnHankock
-== Solution 2.1 (updated with better notation)==
+==Solution 3 (Better notation)==
-Do casework on <math>v_2</math> and <math>v_3.</math>  Notice that we must have <math>v_3(a) = 1</math> and <math>v_2(d)=1</math> and the values of <math>b,d</math> does not matter.  Therefore, <math>\gcd(d,a) = 6k,</math> where <math>k</math> is not divisible by <math>2</math> or <math>3.</math>  We see that <math>13</math> is the only possible answer.
--Williamgolly
+First off, note that <math>24</math>, <math>36</math>, and <math>54</math> are all of the form <math>2^x\times3^y</math>. The prime factorizations are <math>2^3\times 3^1</math>, <math>2^2\times 3^2</math> and <math>2^1\times 3^3</math>, respectively. Now, let <math>a_2</math> and <math>a_3</math> be the number of times <math>2</math> and <math>3</math> go into <math>a</math>, respectively. Define <math>b_2</math>, <math>b_3</math>, <math>c_2</math>, and <math>c_3</math> similarly. Now, translate the <math>lcm</math>s into the following:
+<cmath>1) \min(a_2,b_2)=3</cmath> <cmath>2) \min(a_3,b_3)=1</cmath> <cmath>3) \min(b_2,c_2)=2</cmath> <cmath>4) \min(b_3,c_3)=2</cmath> <cmath>5) \min(c_2,d_2)=1</cmath> <cmath>6) \min(c_3,d_3)=3</cmath>
-==Solution 3 (Better notation)==
+From <math>4)</math>, we see that <math>b_3 \geq 2</math>, thus from <math>2)</math>, <math>a_3 = 1</math>. Similarly, from <math>3)</math>, <math>c_2 \geq 2</math>, thus from <math>5)</math>, <math>d_2 = 1</math>.
+Note also that <math>d_3 \geq 3</math> and <math>a_2 \geq 3</math>. Therefore <cmath>\min(a_2, d_2) = 1</cmath> <cmath>\min(a_3, d_3) = 1</cmath> Thus, <math>\gcd(a, d) = 2 \times 3 \times k</math> for some <math>k</math> having no factors of <math>2</math> or <math>3</math>.
-First off, note that <math>24</math>, <math>36</math>, and <math>54</math> are all of the form <math>2^x\times3^y</math>. The prime factorizations are <math>2^3\times 3^1</math>, <math>2^2\times 3^2</math> and <math>2^1\times 3^3</math>, respectively. Now, let <math>a_2</math> and <math>a_3</math> be the number of times <math>2</math> and <math>3</math> go into <math>a</math>,respectively. Define <math>b_2</math>, <math>b_3</math>, <math>c_2</math>, and <math>c_3</math> similiarly. Now, translate the <math>lcm</math>s into the following:
+Since <math>70 < \gcd(a, d) < 100</math>, the only values for k are <math>12, 13, 14, 15, 16</math>, but all have either factors of <math>2</math> or <math>3</math>, except <math>\boxed{\textbf{(D)} 13}</math>. And we're done.
-<cmath>\min(a_2,b_2)=3</cmath> <cmath>\min(a_3,b_3)=1</cmath> <cmath>\min(b_2,c_2)=2</cmath> <cmath>\min(b_3,c_3)=2</cmath> <cmath>\min(a_2,c_2)=1</cmath> <cmath>\min(a_3,c_3)=3</cmath> .
-(Unfinished)
+~Rowechen Zhong ~MannsNotHot
-~Rowechen Zhong
 ==Solution 4 (Fastest)==
-Notice that <math>gcd (a,b,c,d)=gcd(gcd(a,b),gcd(b,c),gcd(c,d))=gcd(24,36,54)=6</math>, so <math>gcd(d,a)</math> must be a multiple of <math>6</math>. The only answer choice that gives a value between <math>70</math> and <math>100</math> when multiplied by 6 is <math>\boxed{\textbf{(D) } 13}</math>. - mathleticguyyy + einstein
+Notice that <math>\gcd (a,b,c,d)=\gcd(\gcd(a,b),\gcd(b,c),\gcd(c,d))=\gcd(24,36,54)=6</math>, so <math>\gcd(d,a)</math> must be a multiple of <math>6</math>. The only answer choice that gives a value between <math>70</math> and <math>100</math> when multiplied by <math>6</math> is <math>\boxed{\textbf{(D) } 13}</math>. - mathleticguyyy + einstein
-In the case where there can be 2 possible answers, we can do casework on gcd(d,a)
+In the case where there can be 2 possible answers, we can do casework on <math>\gcd(d,a)</math>
 ~Williamgolly
-==Hardness of problem==
+==Solution 5==
+Since <math>\gcd (a,b) = 24</math>, <math>a = 24j</math> and <math>b = 24k</math> for some positive integers <math>j,k</math> such that <math>j</math> and <math>k</math> are relatively prime.
+Similarly , since <math>\gcd (b,c) = 36</math>, we have <math>b = 24k</math> and <math>c=36m</math> with the same criteria. However, since <math>24</math> is not a multiple of <math>36</math>, we must contribute an extra <math>3</math> to <math>b</math> in order to make it a multiple of <math>36</math>. So, <math>k</math> is a multiple of three, and it is relatively prime to <math>m</math>.
+Finally, <math>\gcd (c,d) = 54</math>, so using the same logic, <math>m</math> is a multiple of <math>3</math> and is relatively prime to <math>n</math> where <math>d = 54n</math>.
+Since we can't really do anything with these messy expressions, we should try some sample cases of <math>a,b,c</math> and <math>d</math>. Specifically, we let <math>j = 5, 7, 11, 13</math> or <math>17</math>, and see which one works.
+First we let <math>j= 5</math>. Note that all of these values of <math>j</math> work for the first <math>\gcd</math> expression because they are all not divisible by <math>3</math>.
+Without the loss of generality, we let <math>k=m=3</math> for all of our sample cases. We can also adjust the value of <math>n</math> in <math>d=54n</math>, since there is no fixed value for <math>\gcd(a,d)</math>; there is only a bound.
+So we try to make our bound <math>70 < \gcd(a,d) < 100</math> satisfactory. We do so by letting <math>j=n</math>.
+Testing our first case <math>a=24 \cdot 5</math> and <math>d = 54 \cdot 5</math>, we find that <math>\gcd(a,d) = 30</math>. To simplify our work, we note that <math>\gcd(24,54) = 6</math>, so <math>\gcd(24k, 54k)</math> for all <math>k>1</math> is equal to <math>6k</math>.
+So now, we can easily find our values of <math>\gcd(a,b)</math>:
+<cmath>\gcd(24 \cdot 5, 54 \cdot 5) = 6 \cdot 5 = 30</cmath>
+<cmath>\gcd(24 \cdot 7, 54 \cdot 7) = 6 \cdot 7 = 42</cmath>
+<cmath>\gcd(24 \cdot 11, 54 \cdot 11) = 6 \cdot 11 = 66</cmath>
+<cmath>\boxed{\gcd(24 \cdot 13, 54 \cdot 13) = 6 \cdot 13 = 78}</cmath>
+<cmath>\gcd(24 \cdot 17, 54 \cdot 17) = 6 \cdot 17 = 104</cmath>
+We can clearly see that only <math>j=n=13</math> is in the bound <math>70 < \gcd(a,d) < 100</math>. So, <math>13</math> must be a divisor of <math>a</math>, which is answer choice <math>\boxed{\textbf{(D)}}</math>.
+-FIREDRAGONMATH16
+==Solution 6==
+<asy>
+//Variable Declarations
+defaultpen(0.45);
+size(200pt);
+fontsize(15pt);
+pair X, Y, Z, W;
+real R;
+path quad;
+//Variable Definitions
+R = 1;
+X = R*dir(45);
+Y = R*dir(135);
+Z = R*dir(-135);
+W = R*dir(-45);
+quad = X--Y--Z--W--cycle;
+//Diagram
+draw(quad);
+label("$b$",X,NE);
+label("$a=2^3 \cdot 3 \cdot p$",Y,NW);
+label("$d=2 \cdot 3^3 \cdot q$",Z,SW);
+label("$c$",W,SE);
+label("$2^3 \cdot 3$",X--Y);
+label("$2^2 \cdot 3^2$",X--W);
+label("$2 \cdot 3^3$",Z--W);
+label("$2 \cdot 3 \cdot k$",Z--Y);
+</asy>
+The relationship of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> is shown in the above diagram. <math>gcd(a,d)=2 \cdot 3 \cdot k</math>.
+<math>70 < 6k < 100</math>, <math>12 \le k \le 16</math>, <math>k=\boxed{\textbf{(D) }13}</math>
+Note that <math>gcd(b,c)=36</math> is not required to solve the problem.
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+==Solution 7 (Easier version of Solution 1)==
+Just as in solution <math>1</math>, we prime factorize <math>a, b, c</math> and <math>d</math> to observe that
+<math>a=2^3\cdot{3}\cdot{w}</math>
+<math>b=2^3\cdot{3^2}\cdot{x}</math>
+<math>c=2^2\cdot{3^3}\cdot{y}</math>
+<math>d=2\cdot{3^3}\cdot{z}.</math>
+Substituting these expressions for <math>a</math> and <math>d</math> into the final given,
+<math>70<\text{gcd}(2\cdot{3^3}\cdot{z}, 2^3\cdot{3}\cdot{w})<100.</math>
+The greatest common divisor of these two numbers is already <math>6</math>. If <math>k</math> is what we wish to multiply <math>6</math> by to obtain the gcd of these two numbers, then
-The ranking of this number theory problem is easy hard(just starting to be hard) or a 7 out of 10. This is because you've got to think through out although it is more of which and easy solution
+<math>70<6k<100</math>. Testing the answer choices, only <math>13</math> works for <math>k</math> (in order for the compound inequality to hold). so our gcd is <math>78</math>, which means that <math>\boxed{\textbf{(D) }13}</math> must divide <math>a</math>.
-~justin6688
+-Benedict T (countmath1)
 == Video Solution by Richard Rusczyk ==
@@ Line 67: / Line 157: @@
 ~savannahsolver
-== Video Solution (Meta-Solving Technique) ==
+== Video Solution by OmegaLearn (Meta-Solving Technique) ==
 https://youtu.be/GmUWIXXf_uk?t=1003

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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