MultiplicationPrinciple AIME Basic
2013


Problem - 181

Find the number of five-digit positive integers, $n$, that satisfy the following conditions:

  • the number $n$ is divisible by $5$,
  • the first and last digits of $n$ are equal, and
  • the sum of the digits of $n$ is divisible by $5$.

Show solution

Clearly, the first and the last digits are both $5$. Therefore, as long as the sum of the middle three digits is a multiple of $5$, all the requirements will be met. Given any choices for the second and the third digits, there are exactly two choices for the fourth digit in order to meet the requirement.

  • First digit: $1$ choice
  • Last digit: $1$ choice
  • Second digit: $10$ choices
  • Third digit: $10$ choices
  • Forth digit: $2$ choices

Hence the answer is $$1\times 1\times 10\times 10\times 2= \boxed{200}$$

 

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