Algebra Know-It-ALL

42 Natural Numbers and Integers

The best way to find out whether or not a large odd number is prime is to try to factor it into primes.

If the only factors you get are itself and 1 (i.e., if you can’t factor it into primes), then your number is

prime. There are some other techniques you can use determine when a number is not prime, such as the

“divisibility” tricks you’ll see later in this chapter.

Here’s a challenge!

When an even number is multiplied by 7, the result always even. Show why this is true.

Solution

For the first few even natural numbers, multiplication by 7 always gives you an even number. Here are the

examples for all the single-digit even numbers:

0 × 7 = 0

2 × 7 = 14

4 × 7 = 28

6 × 7 = 42

8 × 7 = 56

You can prove that multiplying any even number by 7 always gives you an even number if you realize

that the last digit of an even number is always even. Think of an even number p—any even number. This

numberp, however large it might be, must look like one of the following:

______0

______2

______4

______6

______8

where the long underscore represents any string of digits you want to put there. Now think of “long mul-

tiplication” by 7. Remember how you arrange the numerals on the paper and then do the calculations.

You always start out by multiplying the last digits of the two numbers together, getting the last digit of the

product. The even number on top, which you are multiplying by the number on the bottom, must end

in 0, 2, 4, 6, or 8. If the number on the bottom is 7, then the last digit in the product must be 0, 4 (the

second digit in 14), 8 (the second digit in 28), 2 (the second digit in 42), or 6 (the second digit in 56)

respectively. The product of any even number and 7 is therefore always even.

Natural Number Nontrivia

Here are some interesting facts about natural numbers. I was about to call them “trivia,”

but after thinking about it for awhile, I decided that the ones involving primes are not

trivial at all!