# Tag Archives: modulo arithmetic

## 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

## Remainder of Division

What is the remainder when is divided by ? Source: NCTM Mathematics Teacher, August 2006 Solution We arrange all the non-negative integers into buckets depending on the value of the remainder when they are divided by . It works the … Continue reading

## Divided by 5

Find the reminder when is divided by ? Source: NCTM Mathematics Teacher, October 2006 Solution is such a large number that it is impractical to use modulo arithmetic to find the remainder. So we are going to find the remainders … 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

## Pairs of Digits

How many pairs of decimal digits exist with the property that if is an integer ending in those two digits, then ends in the same two digits? Repetition of the digits in a pair is allowed. Source: NCTM Mathematics Teacher, … Continue reading

## Sum of Powers of Five

Compute the units digit of . Source: NCTM Mathematics Teacher 2008 SOLUTION 1 The units digits of follows the following pattern. If , the units digits of the sum are . For example, , units digit = , units digit … 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

## Hill Cipher

I. Consider the following equation where and represent two relatively prime integers (a) Verify that the ordered pair is a solution of Eq. (b) Use part (a) to find a parametric solution of Eq. (c) Derive an integer such that … Continue reading

## No Integer Solutions

1. Show that the following system of equations has no integer solutions 2. Show that the following system of equations has no integer solutions Source: http://www.math.rutgers.edu/~erowland/modulararithmetic.html SOLUTION 1. We reduce the equations and hope to find a contradiction thereby showing … Continue reading