## Difference of Sequences

For the sequence , what is the difference between the mean of the sequence’s first terms and the mean of its first terms? Source: NCTM Mathematics Teacher February 2008 Solution If we add the first terms by pairs we end … Continue reading

## 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

## 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

## 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

## 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

## 4-Digit Numbers Divisible by 7

Given the digits , and , find the probability that a four-digit number formed by using each of them only once is divisible by . Source: NCTM Mathematics Teacher, February 2008 Solution There are possible four-digit numbers using each only … Continue reading

## Two-mile Long Train

How long will it take a two-mile long train traveling miles per hour to travel completely through a mile-long tunnel? Source: NCTM Mathematics Teacher, February 2008 Solution The diagram shows that for the train to clear the tunnel, point must … Continue reading

## 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”.