2001 AMC 8 Problems/Problem 25

Problem

There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?

$\textbf{(A)}\ 5724 \qquad \textbf{(B)}\ 7245 \qquad \textbf{(C)}\ 7254 \qquad \textbf{(D)}\ 7425 \qquad \textbf{(E)}\ 7542$

Solution

We begin by narrowing down the possibilities. If the larger number were twice the smaller number, then the smallest possibility for the larger number is $2457\times2=4914$ , since $2457$ is the smallest number in the set. The largest possibility would have to be twice the largest number in the set such that when it is multiplied by $2$ , it is less than or equal to $7542$ , the largest number in the set. This happens to be $2754\times2=5508$ . Therefore, the number would have to be between $4914$ and $5508$ , and also even. The only even numbers in the set and in this range are $5472$ and $5274$ . A quick check reveals that neither of these numbers is twice a number in the set. The number can't be quadruple or more another number in the set since $2457\times4=9828$ , well past the range of the set. Therefore, the number must be triple another number in the set. The least possibility is $2457\times3=7371$ and the greatest is $2475\times3=7425$ , since any higher number in the set multiplied by $3$ would be out of the range of the set. Reviewing, we find that the upper bound does in fact work, so the multiple is $2475\times3=7425, \boxed{\textbf{(D)}}$

Or, since the greatest number possible divided by the smallest number possible is slightly greater than 3, divide all of the choices by 2, then 3 and see if the resulting answer contains 2,4,5 and 7. Doing so, you find that $7425/3 = 2475, \boxed{\textbf{(D)}}$

Solution 2

There are only 5 options for the problem so we can just try them. It is easy since that we only need try to use $2$ , $3$ to divide them. Even $5$ will leads to a solution start with $1$ which we don't need.

$5724/2=2862$ , $5724/3=1908$ . $7245/3=2415$ . $7254/2=3612$ , $7254/3=2418$ . $7425/3=2475$ . The answer is $\boxed{\textbf{(D)}}$ . You can obtain the answer in only 6 calculations.

2001 AMC 8 (Problems • Answer Key • Resources)
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2001 AMC 8 Problems/Problem 25

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Problem

Solution

Solution 2

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