2010 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra Number Theory

A palindrome, such as

83438,

is a number that remains the same when its digits are reversed. The numbers

x

and

x + 32

are three-digit and four-digit palindromes, respectively. What is the sum of the digits of

x?

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Solution

Note that

x

is at most

999.

This means that

x + 32

has a maximum of

1031.

Similarly, we have that the minimum value of

x + 32

1000.

The only palindrome in this range is

1001,

so this is what

x + 32

equals. Thus

x=1001-32=969,

whose digit sum is

9+6+9=24.

Thus, E is the correct answer.