## Number Theory and Averages

When a × b = c is true, then a and b are factors of c and c is divisible by a and b. Being able to tell whether one number will divide another number with no remainder is a useful skill. It can be used to check whether a division problem will have a remainder and also to find the factors of a number.

At this grade level, divisibility tests for 2, 5, and 10 are developed. All even numbers are divisible by 2. Odd numbers are not divisible by 2. Since all even numbers have 0, 2, 4, 6, or 8 as the last digit, all numbers ending in 0, 2, 4, 6, or 8 are divisible by 2. A number is divisible by 5 if and only if the last digit is 0 or 5. A number is divisible by 10 if and only if the last digit is 0.

A number is prime if it has exactly two factors, 1 and the number itself. The first seven prime numbers are 2, 3, 5, 7, 11, 13, and 17. Teachers and students would do well to commit these numbers to memory. Notice that 1 is not a prime number because it has only one factor. A number greater than 1 that is not prime is said to be composite. A composite number has more than two factors. The number 6 is composite because its factors are 1, 2, 3, and 6.

**Finding the Mean**

One application of division is finding the average, or mean, of a group of numbers. The average is found by adding the numbers and then dividing by the number of addends. The average tells what each number would be if the sum remained the same and the individual numbers were equal.

Example: Find the average for 9, 19, 44, 8, 25, 4, 27, and 32.

Divide the sum by

the number of addends.

168 ÷ 8 = 21

The average, or mean, of the group is 21.

The average is only one measure of central tendency of a set of data. Other measures of central tendency will be studied later.

**Teaching Model 10.1:** Factors and Multiples