Tag Archives: multiple

Difference of Two Squares

Which one of the following numbers cannot be expressed as the difference of the squares of two integers? (a) (b) (c) (d) (e) Source: NCTM Mathematics Teacher, September 2006 Solution Let be an odd integer and for some integer and … Continue reading

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Values of Integers

If and are integers such that , what are the possible values of ? (a) only (b) only (c) only multiples of (d) any integer (e) none of these Source: NCTM Mathematics Teacher, September 2006 Solution Let be an integer … Continue reading

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TV Game Show

On a television show, a player receives points for answering an easy question and points for a hard one. What is the largest integer that cannot be a contestant’s total score of the game? Source: NCTM Mathematics Teacher, October 2006 … Continue reading

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Multiple of 3

If is an  integer, show why must be a multiple of . Source: NCTM Mathematics Teacher, October 2006 Solution which is a rational number. Clearly, for to be an integer must be a multiple of . Answer: Shown in solution

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Largest of Five Integers

A set contains five integers. When distinct elements of this set are added together, two at a time, the complete list of different possible sums is , and . What is the largest of the five integers in the set? … Continue reading

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Perfect Squares and Perfect Cubes

List all integers less than that are both perfect squares and perfect cubes. Source: NCTM Mathematics Teacher, January 2006 Solution is a perfect square but not a perfect cube. On the other hand, is a perfect cube but not a … Continue reading

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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

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Chicken Nuggets

Chicken nuggets come in packets of , or . What is the largest number of nuggets that you cannot buy when combining various packets? Source: NCTM Mathematics Teacher 2006 SOLUTION Consider the numbers , and All even numbers greater than … Continue reading

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Fives and Eights

Find the greatest positive integer that cannot be expressed as a multiple of , as a multiple of , or as a sum of a multiple of and a multiple of . Source: mathcontest.olemiss.edu 3/24/2014 SOLUTION What kind of numbers satisfy the … Continue reading

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Sum of Four Consecutive Integers

How many positive multiples of that are less than are the sum of four consecutive integers? Source: NCTM Mathematics Teacher SOLUTION The pattern is These multiples of when plotted on a number line are separated by an interval of . … Continue reading

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