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

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

Revision as of 00:36, 5 December 2019

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 divdes $w$, 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' $2^3 * 3$, that $b$ and $c$ have $2^2 * 3^2$, and that $c$ and $d$ have $3^3 * 2$. Combining $1$ and $2$ yields $b$ has (at a minimum) $2^3 * 3^2$, and thus $a$ has $2^3 * 3$ (and no more powers of $3$ because otherwise $gcd(a,b)$ would be different). In addition, $c$ has $3^3 * 2^2$, and thus $d$ has $3^3 * 2$ (similar to $a$, we see that $d$ cannot have any other powers of $2$). We now assume the simplest scenario, where $a = 2^3 * 3$ and $d = 3^3 * 2$. According to this base case, we have $gcd(a, d) = 2 * 3 = 6$. 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 $6 * 13$ is the only one that works. Therefore the answer is $\boxed{\textbf{(D) } 13}$

Solution by JohnHankock

Solution 2.1

Elaborating on to what Solution 1 stated, we are not able to add any extra factor of 2 or 3 to $gcd(a, d)$ because doing so would later the $gcd$ of $(a, b)$ and $(c, d)$. This is why:

The $gcd(a, b)$ is $2^3 * 3$ and the $gcd$ of $(c, d)$ is $2 * 3^3$. However, the $gcd$ of $(b, c) = 2^2 * 3^2$ (meaning both are divisible by 36). Therefore, $a$ is only divisible by $3^1$ (and no higher power of 3), while $d$ is divisible by only $2^1$ (and no higher power of 2).

Thus, the $gcd$ of $(a, d)$ can be expressed in the form $2 * 3 * k$ for which $k$ is a number not divisible by $2$ or $3$. The only answer choice that satisfies this (and the other condition) is $\boxed{\textbf{(D) } 13}$.

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 $c_3$ similiarly. Now, translate the $lcm$s into the following: \[\min(a_2,b_2)=3\] \[\min(a_3,b_3)=1\] \[\min(b_2,c_2)=2\] \[\min(b_3,c_3)=2\] \[\min(a_2,c_2)=1\] \[\min(a_3,c_3)=3\] .

(Unfinished) ~Rowechen Zhong

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

See Also

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

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