Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Prove it Philippines

Problem:

Let $A$ , $B$ , $C$ be positive integers such that the number $1212017ABC$ is divisible by $45$ . Find the difference between the largest and the smallest possible values of the two-digit number $AB$ .

Solution

Solution:

Since the number is divisible by $45$ , it must be divisible by both $9$ and $5$ .

If $C$ is $0$ , then $A$ and $B$ must have a sum of either $4$ or $13$ .

If $C$ is $5$ , then $A$ and $B$ can either have a sum of $8$ or $17$ .

Based from these, the largest and smallest values of $AB$ are $98$ and $13$ , respectively, which have a difference of $85$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.