Tag Archives: number sense

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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Divisible by Units Digit

The number is divisible by its units digit . How many whole numbers less than are divisible by their respective units digit? Source: NCTM Mathematics Teacher, February 2008 Solution Nine numbers: divisible by themselves Six numbers: divisible by Six numbers: … Continue reading

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Sum of 7’s

In the sum of the expression , what digit will be in the tens place? Source: NCTM Mathematics Teacher, February 2008 Solution We are adding nineteen numbers from the smallest to the largest . Note that there are nineteen in … Continue reading

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Numerical Puzzle

The five-square numerical puzzle below requires the vertical column to be filled with the digits from a three-digit integral power of and the horizontal row to be filled with the digits from a three-digit integral power of . What digit … Continue reading

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Triangular Array of Numbers

If you continue the triangular array of numbers shown in the figure, what number would be directly below ? Source: NCTM Mathematics Teacher, February 2008 Solution Note that the leading numbers increase by , etc. If we follow this pattern, … Continue reading

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Smaller Factor

Given a three-digit number, interchange the units digit and the hundreds digit. The product of the original number and the new number is . What is the smaller three-digit factor? Source: NCTM Mathematics Teacher, February 2008 Solution Since the product … Continue reading

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Square of 111,111,111

Find the exact value of . Source: NCTM Mathematics Teacher, February 2008 Solution Answer: Alternative solution Multiply the numbers manually using the old-fashioned multiplication algorithm. It should be straightforward because there is no “carry”.

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Two Even Digits

The number is a four-digit number having two zeros and two even digits that are not zeros. How many four-digit numbers besides can also be described this way? Source: NCTM Mathematics Teacher, February 2008 Solution First, choose a way to … Continue reading

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Product of Factorials

Solve for if . Source: NCTM Mathematics Teacher, February 2008 Solution Answer:

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Three Consecutive Odd Numbers

Find three consecutive odd numbers such that the sum of their squares is a four-digit number with all four digits the same. Source: NCTM Mathematics Teacher, February 2008 Solution The squares of odd numbers are odd and the sum of … Continue reading

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