# Tag Archives: divisibility

## Divide Evenly

Consider the set of positive integers for which there exists an integer such that evenly divides both and . Find the sum of the elements of . Source: NCTM Mathematics Teacher, August 2006 Solution The integers that are divided evenly … Continue reading

## Divisible by 7

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 … Continue reading

## Divisible by 25

How many 5-digit numbers with all digits nonzero and no digit repeated are divisible by ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION A number is divisible by if it ends in , or . The nonzero digit condition restricts … Continue reading

## ISBN

Commercially published books bear a 10-digit identification number called an ISBN. Let denote the digits from left to right. Each ISBN has the property that is divisible by . Given the correct ISBN , how many possible pairs of digits … Continue reading

## Lucas Sequence

Consider the so-called Lucas sequence: , in which every number after the second is the sum of the two preceding numbers. Prove that not a single member of this sequence is divisible by . Source: NCTM Mathematics Teacher SOLUTION The … Continue reading

## Divisibility of Pythagorean Triples

1. Show that if are integers such that , then (a) at least one of them is divisible by (b) at least one of them is divisible by (c) at least one of them is divisible by . 2. Show … Continue reading