2014 AMC 12A Problem 6

Problem 6 of 25EasierNumber Theory

The difference between a two-digit number and the number obtained by reversing its digits is

5

times the sum of the digits of either number. What is the sum of the two-digit number and its reverse?

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Solution

Let the larger number be

10a+b.

Then

\begin{aligned} &(10a+b)-(10b+a)=9(a-b)\\ &\quad{}=5(a+b), \end{aligned}

which simplifies to

2a=7b.

The only nonzero digits satisfying this are

a=7

and

b=2,

so the number is

72

and its reverse is

27.

Their sum is

72+27=99.

Thus, the correct answer is D.