Tag Archives: number sense

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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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

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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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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

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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

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Sum of Three Integers

For certain positive integers , and , . If and are prime, find . Source: SCVMA Math Olympiad 2010 Solution Since and are odd, and are odd which implies that is even. Hence is even because the sum is even. … Continue reading

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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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Product of Consecutive Integers

How many integer pairs satisfy the equation ? Source: NCTM Mathematics Teacher, September 2006 Solution We construct products of positive consecutive integers and see which ones if any can be expressed as a power of . is the only such … 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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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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