2022 CEMC Cayley Problems/Problem 2

Problem

The integer is a multiple of

Solution 1

First, we can see that the number is not divisible by nor because the last digit is not divisible by either, according to the division rule for and . This eliminates answer choices A and C.

For the number to be divisible by , the sum of the digits must be a multiple of . , which is not divisible by . This eliminates answer choice B.

A number is also divisible by if and only if the result of subtracting the odd numbered digits from the even digits is divisible by . We have .

This is not divisible by , so answer choice E is eliminated.

A, B, C, and E are eliminated. Thus, the only answer choice remaining is .

~anabel.disher

Solution 2

We can remember that the prime factorization of is . Thus, the factors of are , , , and .

Of these factors, the only one that is an answer choice is .

~anabel.disher