Source: NCTM Mathematics Teacher February 2008

**Solution**

If we add the first terms by pairs we end up with . The mean of the first terms is .

Likewise, if we add the first terms by pairs, we get . The mean of the first terms is .

The difference between the two means is .

**Answer**:

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Source: NCTM Mathematics Teacher, February 2008

**Solution**

Given that , the number of divisors is . The first twelve divisors are small and easy to guess: , and and as a bonus we get the last twelve by dividing by the first twelve divisors. For example, , etc. : .

When we add the reciprocals of the divisors, the divisors appear as denominators in a set of 24 fractions

When we reduce the fractions to (the least common denominator), the divisors appear as numerators

**Answer**:

Source: NCTM Mathematics Teacher, February 2008

**Solution**

Nine numbers: divisible by themselves

Six numbers: divisible by

Six numbers: divisible by

Five numbers: divisible by

Six numbers: divisible by their respective units digit

Total

**Answer**:

Source: NCTM Mathematics Teacher, February 2008

**Solution**

We are adding nineteen numbers from the smallest to the largest . Note that there are nineteen in the ones place and eighteen in the tens place. First, we add the nineteen in the ones place and get . Consider the number . The in the ones place of will be the ones place digit of the final sum. The in the tens place of means and if we regroup with the eighteen , we get . The digit in the tens place of the final sum is .

**Answer**:

Source: NCTM Mathematics Teacher, February 2008 ]]>

Source: NCTM Mathematics Teacher, February 2008

**Solution**

There are possible four-digit numbers using each only once

Numbers divisible by are

Probability = .

**Answer**:

Source: NCTM Mathematics Teacher, February 2008

**Solution**

The diagram shows that for the train to clear the tunnel, point must travel three mile-long segments each of which takes hour or minutes to finish. The train will clear the tunnel in minutes.

**Answer**: minutes

**Alternative solution**

When the head of the train enters the tunnel, the end of the train is miles away from the exit. The rear of the train will have to travel miles at miles per hour.

Source: NCTM Mathematics Teacher, February 2008

**Solution**

Note that the leading numbers increase by , etc. If we follow this pattern, the leading numbers of the first rows are , which means that the number directly below is .

**Answer**:

**Alternative solution 1**

Note that the ending number of each row is a perfect square , etc. If we follow this pattern, is the ending number of row and is the ending number of row .

**Alternative solution 2**

Let represent the row number. We want to find a function that relates row to the leading number of that row. We already know a few values of

Is a linear or quadratic function or neither? The first differences (FD) and second differences (SD) in the values of will tell us what it is

is a quadratic function because the second differences are a constant

because the term does not contribute anything to the function

Source: NCTM Mathematics Teacher, February 2008

**Solution**

Since the product of the original number and the new number is odd, the two numbers must be odd. Furthermore, their units digits must multiply to . These restrictions limit the possibilities to , and . Let’s examine a few numbers to see if we can narrow down the choices starting with

– not possible, too big a product

– not possible

– not possible

The smaller three-digit factor is .

**Answer**:

**Alternative solution**

The prime factorization of .

is a prime number and any multiple of is a -digit number, thus the two factors must be and .

Source: NCTM Mathematics Teacher, February 2008

**Solution**

triangles on the faces of the cube have side lengths of .

triangles formed by two diagonals and one side have side lengths of .

triangles that are bases of tetrahedrons formed by connecting three corners of the cube have side lengths of .

There are distinct (non-congruent) triangles.

**Answer**: