Divisibility

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DIVISIBILITY

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The concepts for this exercise are summarized below; for a complete discussion, read the text.

The numbers 0, 2, 4, 6, … are called even numbers.
Even numbers always end in 0, 2, 4, 6, or 8.
Even numbers can always be divided into two equal (even) piles.
Note: The numbers 1, 3, 5, 7, … are called odd numbers.

The numbers 0, 2, 4, 6, … are also said to be divisible by 2.
Divisible by 2 means that 2 goes into the number evenly.
The phrases even and divisible by 2 are interchangeable.

Divisible by 3 means that 3 goes into the number evenly.
Divisible by 4 means that 4 goes into the number evenly; and so on.

A divisibility test is a shortcut to decide if a number is divisible by a given number.

DIVISIBILITY BY 2
If a number ends in 0, 2, 4, 6, or 8, then the number is divisible by 2.
Also, if a number is divisible by 2, then it ends in 0, 2, 4, 6, or 8.

For example, 87,356 is divisible by 2, since it ends in the digit 6.
However, 87,357 is not divisible by 2, since it ends in the digit 7.

There's a neat trick for deciding if a number is divisible by 3.
The technique is illustrated with the following example:
Is 57,394 divisible by 3? (Answer without using your calculator!)
Add up the digits in the number: 5 + 7 + 3 + 9 + 4 = 28 .
Add up the digits again: 2 + 8 = 10 .
Since 10 is not divisible by 3, the original number 57,394 is also not divisible by 3.

DIVISIBILITY BY 3
To decide if a number is divisible by 3, add up the digits in the number.
Continue this process of adding the digits until you get a manageable number.
(If you want, keep going until you get a single-digit number.)
If this final number is divisible by 3, then the number you started with is also divisible by 3.
If this final number is not divisible by 3, then the number you started with is not divisible by 3.

Read the text for a proof of the "divisibility by 3" test.

DIVISIBILITY BY 5
If a number ends in 0 or 5, then the number is divisible by 5.
Also, if a number is divisible by 5, then it ends in 0 or 5.

Read the text for a clever "finger trick" for divisibility by 9.

DIVISIBILITY BY 10
If a number ends in 0, then the number is divisible by 10.
Also, if a number is divisible by 10, then it ends in 0.

More compactly, we can say that a number is divisible by 10 if and only if it ends with a 0 .

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