Tag Archives: multiple

Divide Evenly

Consider the set of positive integers for which there exists an integer such that evenly divides both and . Find the sum of the elements of . Source: NCTM Mathematics Teacher, August 2006 Solution The integers that are divided evenly … Continue reading

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

Remainder of Sum of Factorials

What is the remainder when is divided by ? Source: NCTM Mathematics Teacher, August 2006 Solution divided by leaves a remainder of . is a multiple of so will have the same remainder as . Likewise, , etc. are multiples … Continue reading

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

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