# Tag Archives: prime factorization

## Sums of Reciprocals

The sum of the positive divisors of is . Find the sum of the reciprocals of the positive divisors of . Source: NCTM Mathematics Teacher, February 2008 Solution Given that , the number of divisors is . The first twelve … Continue reading

## Smaller Factor

Given a three-digit number, interchange the units digit and the hundreds digit. The product of the original number and the new number is . What is the smaller three-digit factor? Source: NCTM Mathematics Teacher, February 2008 Solution Since the product … Continue reading

## Set of Primes

Let be the set of primes that divide . What is the largest integer so that the set of primes that divides is equal to ? Source: NCTM Mathematics Teacher, August 2006 Solution The next prime greater than is . … Continue reading

## Partners in Integers

Two integers are said to be partners if both are divisible by the same set of prime numbers. Find the number of positive integer less than that have no partners less than . Source: NCTM Mathematics Teacher, August 2006 Solution … Continue reading

## Product of Consecutive Integers

How many integer pairs satisfy the equation ? Source: NCTM Mathematics Teacher, September 2006 Solution We construct products of positive consecutive integers and see which ones if any can be expressed as a power of . is the only such … Continue reading

## Cute Number

A number is called cute if it has exactly four positive integral divisors. What percent of the first twenty-five positive integers are cute? Source: NCTM Mathematics Teacher, January 2006 SOLUTION number of factors/divisors Other integers that have factors/divisors: . Total number … Continue reading

## Prime Factors

Find the prime factors of . Source: NCTM Mathematics Teacher, January 2006 SOLUTION Recall that Both and are prime numbers. The prime factors of are and . Answer: and

## Perfect Square

Find the smallest positive integer such that is a perfect square, is a perfect cube, and is a perfect fifth power. Source: NCTM Mathematics Teacher, December 2005 SOLUTION If is a perfect square, must have as a prime factor such … Continue reading

## Sum of Consecutive Numbers (Part 2)

The sum of consecutive positive integers is . Find all possible values of . Source: NCTM Mathematics Teacher, December 2005 SOLUTION Let be the sum of consecutive positive integers with the first term equal . If we add the first … Continue reading

## Pennies on Table

Some pennies are piled on a table. If you put them into piles of size , or , you always end up with penny left over. There is more than penny on the table. What is the fewest number of … Continue reading