Math CentralQuandaries & Queries


Question from Daniel:

How many palindromes of 5 digits are even?

Hi Daniel,

A positive integer is even if its units digit is even and conversely if a positive integer is even its units digit is even. Hence for a 5 digit positive integer to be even its units digit must be 0, 2, 4, 6 or 8. If this 5 digit integer is a palindrome its ten thousands digit must be the same as its units digit. But if the units digit is 0 then so is its ten thousands digit and it is only a 4 digit number. Hence a 5 digit even palindrome must have a units digit of 2, 4, 6 or 8, four possibilities.

For each possible choice of a units digit the tens digit can be any digit, hence ten possibilities. Once the tens digit is chosen the fact that it is a palindrome the thousands digit is fixed.

How many choices are there for the hundreds digit?


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