How many four digit positive integers divisible by have the property that, when the first and last digits are interchanged, the result is a (not necessarily four-digit) positive integer divisible by ?

Source: NCTM Mathematics Teacher, October 2006

**Solution
**Is divisible by ? One way to find out is by determining if “behaves” like or or some other small multiples of . Modulo arithmetic will help us do that. In its most basic definition, modulo arithmetic is the arithmetic of remainder. We say “ is congruent to modulo ” and write , because and yield the same remainder when divided by .

Using , we will do the same calculation but with congruent numbers mod . For ease in presentation we drop the modulo notation.

Since ,

is divisible by .

Let be a 4-digit positive integer where the digits are , and . Suppose divides and . Show that divides .

Since divides and , there exist integers and such that

Subtract Eq. from Eq.

where

In other words, divides . Since does not divide , must divide .

Show that if divides , then divides .

Given for some integer ,

Since divides and , must divide . Hence, divides .

We have two cases to consider: and .

Case 1:

The integers are of the form where the numbers are divisible by . The possible numbers are: . The possible digits are: . There are possible .

Cases 2:

The integers are of the form where digits and are congruent. Interchanging congruent digits like and , and , and has no bearing on the remainder. The possible are: . There are possible .

Total number of integers equals .

List of some of the integers

**Answer**: