# Tag Archives: factors

## Three Numbers in Geometric Progression

If the product of three numbers in geometric progression is and their sum is , find the largest of the three numbers. Source: NCTM Mathematics Teacher, August 2006 Solution We assume that the three numbers are not consecutive but in … 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

## 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

## Integral Factors

What is the total number of positive integral factors of ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Integral factors are made by multiplying the powers of and . We can choose from powers of from to , powers of … Continue reading

## Black and White Marbles

Each of two boxes contains marbles, and each marble is either black or white. The total number of black marbles is different from the total number of white marbles. One marble is drawn at random from each box. The probability … Continue reading

## Factors of Polynomial

How many different polynomials of the form with rational coefficients are factors of the polynomial ? Source: NCTM Mathematics Teacher, February 2006 SOLUTION Putting all the factors together into Eq. There are ways to get a product of degree from the factors … Continue reading

## Arithmetic sequence

The terms form an arithmetic sequence whose sum is . The terms in that order, form a geometric sequence. Find the sum of all possible values for . Source: NCTM Mathematics Teacher, February 2006 SOLUTION Let represent the common difference … Continue reading

## XYZ=4000

How many ordered triples of positive integers satisfy ? Source: NCTM Mathematics Teacher 2006 SOLUTION This problem is similar to the problem “Triple ABC” posted on 9/14/2012, nevertheless a different solution is offered here. Given that , the positive integers … Continue reading

## 2008 As Sum of Consecutive Integers

The number can be written as a sum of consecutive integers such that the number of terms in the sum is twice the number of factors of . Find the smallest of these consecutive integers. Source: NCTM Mathematics Teacher 2008 … Continue reading