1957 AHSME Problems/Problem 38

Problem

From a two-digit number $N$ we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then:

$\textbf{(A)}\ {N}\text{ cannot end in 5}\qquad\\ \textbf{(B)}\ {N}\text{ can end in any digit other than 5}\qquad \\ \textbf{(C)}\ {N}\text{ does not exist}\qquad\\ \textbf{(D)}\ \text{there are exactly 7 values for }{N}\qquad\\ \textbf{(E)}\ \text{there are exactly 10 values for }{N}$

Solution

The number $N$ can be written as $10a+b$ with $a$ and $b$ representing the digits. The number $N$ with its digits reversed is $10b+a$ . Since the problem asks for a positive number as the difference of these two numbers, than $a>b$ . Writing this out, we get $10a+b-(10b+a)=9a-9b=9(a-b)$ . Therefore, the difference must be a multiple of $9$ , and the only perfect cube with less than $3$ digits and is multiple of $9$ is $3^3=27$ . Also, that means $a-b=3$ , and there are $7$ possibilities of that, so our answer is $\boxed{\textbf{(D) } \text{There are exactly } 7 \text{ values for } N}$ .

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 37	Followed by Problem 39
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1957 AHSME Problems/Problem 38

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