Tag Archives: multiple

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

Posted in Problem solving | Tagged , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , | Leave a comment

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

Posted in Problem solving | Tagged , , , , , , , , , , | Leave a comment