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