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Divisibility and Primes
Factors and Multiples
Divisibility Rules
Prime Numbers
The Distribution of Primes
Lowest Common Multiples
Greatest Common Factors

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Divisibility and PrimesFactors and Multiples

読書の時間: ~15 min

By now you should be comfortable with addition, subtraction and multiplication of integers. Division is slightly different, because you can’t always divide any integer by any other. For example 17 divided by 3 is not a whole number – it is somewhere in between 5 and 6. You either have to give a remainder (2), or express the answer as a decimal number (5.66…).

0 1 2 3 4 5 6 7 8 9 10 11 12 3 3 12 3 3

12 is divisible by 3

0 1 2 3 4 5 6 7 8 9 10 11 12 4 4 4 10

10 is not divisible by 4

If you can divide a number A by a number B, without remainder, we say that B is a factor (or divisor) of A, and that A is a multiple of B. We often write B|A, where the vertical bar simply means “divides”.

For example, 7 × 3 = 21, so 7 is a of 21. Similarly, 21 is a of 7, and we can write 7|21.

In this short game you have to determine which numbers are factors or multiples:

Factors and Multiples Quiz

${x}
is a
factor
multiple
neither
of
${y}

It is often useful to find all the factors of a number. For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Of course, you don’t want to check all numbers up to 60 if they are factors. Instead, there is a simple technique which relies on the fact that factors always appear in .

In the case of 60 we have 60 = 1 × 60 = 2 × 30 = 3 × 20 = 4 × 15 = 5 × 12 = 6 × 10. Or, in a different notation,

601,2,3,4,5,6,10,12,15,20,30,60

To find all factors of a number we simply start at both ends of this list, until we meet in the middle.

421,2,3,6,7,14,21,42
For example, the first factor pair of 42 is simply 1 and 42, and we write them down with much space in between.
After 1 at the beginning, we check if 2 divides 42. It does, and the corresponding pair is 42 ÷ 2 = 21.
Next, we check if 3 divides 42. It does, and the corresponding pair if 42 ÷ 3 = 14.
Now we check if 4 divides 42. It does not, however, so we move on.
5 also doesn’t divide 42 so we move on.
6 does divide 42 again. Its pair is 42 ÷ 6 = 7. Notice how we’ve met in the middle after only a few attempts, without having to test all numbers from 7 to 42.

The only special case with this method is for square numbers: in that case, you will meet at just a single number in the middle, like 64 = 8 × 8.

Archie