# Category Archives: Problem solving

## 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

## Consecutive Remainders

What is the smallest positive integer that when divided by , and leaves the remainders , and , respectively? Source: NCTM Mathematics Teacher, August 2006 Solution If leaves a remainder of , the possible values of are . The integers … Continue reading

## Divisible by 3 or 7

Find the number of four-digit positive integers divisible by or . Source: NCTM Mathematics Teacher, August 2006 Solution Consider , the set of four-digit positive integers divisible by and , the set of four-digit positive integers divisible by . The … Continue reading

## Area of Trapezoid

In a trapezoid with parallel to , the diagonals intersect at point . The area of triangle is , and the area of triangle is . Find the area of the trapezoid. Source: NCTM Mathematics Teacher, August 2006 Solution Draw … Continue reading

## Values From Three Coins

How many possible values can there be for three coins selected from among pennies, nickels, dimes, and quarters. Source: NCTM Mathematics Teacher, August 2006 Solution a) All three coins are the same Number of ways = b) Two of the … 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

## 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

## Largest 4-digit Integer x

Consider the equation . Find the largest four-digit integer for which there is an integer so that the pair is a solution. Source: NCTM Mathematics Teacher, August 2006 Solution For to be an integer, must be a multiple of . … Continue reading

## Two Defective Tiles

An -by–ft area has been tiled with -foot-square tiles. Two of the tiles were defective. What is the probability that the two defective tile share an edge? Source: NCTM Mathematics Teacher, August 2006 Solution Area Vertical shared edges Horizontal shared … Continue reading

## Choosing an Integer

An experiment consists of choosing with replacement an integer at random among the numbers from to inclusive. If we let denote a number that is an integral multiple of and denote a number that is not an integral multiple of … Continue reading