Tag Archives: divisor

Sums of Reciprocals

The sum of the positive divisors of is . Find the sum of the reciprocals of the positive divisors of . Source: NCTM Mathematics Teacher, February 2008 Solution Given that , the number of divisors is . The first twelve … 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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Cute Number

A number is called cute if it has exactly four positive integral divisors. What percent of the first twenty-five positive integers are cute? Source: NCTM Mathematics Teacher, January 2006 SOLUTION number of factors/divisors Other integers that have factors/divisors: . Total number … 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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Multiple of 72

The seven-digit integer is a multiple of (where are digits). Determine all possible ordered pairs . Source: NCTM Mathematics Teacher, December 2005 SOLUTION Since and , we multiply by , etc. and by , etc. to find the possible digits … Continue reading

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Find All Integers

Find all integers such that is also an integer. Source: NCTM Mathematics Teacher, December 2005 SOLUTION The expression is an integer if is a divisor of . The divisors of are: Eight integers make the expression also an integer. Answer: 8

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Integral Points

A point is called integral if both and are integers. How many points on the graph of are integral points? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Divide by , Divide by Divide by , no solution , duplicate solution … Continue reading

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More Remainders

How many positive integers have a remainder when is divided by ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION If divided by has a remainder , there exists a quotient such that and . which implies that the number of … Continue reading

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Prime Number

What is the last digit of the 9-digit number , given that the number is prime? Source: NCTM Mathematics Teacher, November 2006 SOLUTION The last digit could be , or . cannot be because . cannot be because . cannot be … Continue reading

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XYZ=4000

How many ordered triples of positive integers satisfy ? Source: NCTM Mathematics Teacher 2006 SOLUTION This problem is similar to the problem “Triple ABC” posted on 9/14/2012, nevertheless a different solution is offered here. Given that , the positive integers … Continue reading

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