Tag Archives: remainder

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

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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

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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

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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

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Divided by 5

Find the reminder when is divided by ? Source: NCTM Mathematics Teacher, October 2006 Solution is such a large number that it is impractical to use modulo arithmetic to find the remainder. So we are going to find the remainders … Continue reading

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Divisible by 7

How many four digit positive integers divisible by have the property that, when the first and last digits are interchanged, the result is a (not necessarily four-digit) positive integer divisible by ? Source: NCTM Mathematics Teacher, October 2006 Solution Is … 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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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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Largest Divisor

Select any prime number greater than . Square it and subtract . What is the largest number that must be a divisor of the result? Source: NCTM Mathematics Teacher, January 2006 SOLUTION Suppose is a prime number greater than . … Continue reading

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Pennies on Table

Some pennies are piled on a table. If you put them into piles of size , or , you always end up with penny left over. There is more than penny on the table. What is the fewest number of … Continue reading

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