Using Common Divisibility Tests
Using Common Divisibility Tests
Another way to say that \(375\) is a multiple of \(5\) is to say that \(375\) is divisible by \(5.\) In fact, \(375÷5\) is \(75,\) so \(375\) is \(5\cdot 75.\) Notice in the last example of the previous lesson that \(10,519\) is not a multiple \(3.\) When we divided \(10,519\) by \(3\) we did not get a counting number, so \(10,519\) is not divisible by \(3.\)
Divisibility
If a number \(m\) is a multiple of \(n,\) then we say that \(m\) is divisible by \(n.\)
Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. The table summarizes divisibility tests for some of the counting numbers between one and ten.
| Divisibility Tests | |
|---|---|
| A number is divisible by | |
| \(2\) | if the last digit is \(0,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}4,\phantom{\rule{0.2em}{0ex}}6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8\) |
| \(3\) | if the sum of the digits is divisible by \(3\) |
| \(5\) | if the last digit is \(5\) or \(0\) |
| \(6\) | if divisible by both \(2\) and \(3\) |
| \(10\) | if the last digit is \(0\) |
Example
Determine whether \(1,290\) is divisible by \(2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.\)
Solution
The table applies the divisibility tests to \(1,290.\) In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.| Divisible by…? | Test | Divisible? | Check |
|---|---|---|---|
| \(2\) | Is last digit \(0,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}4,\phantom{\rule{0.2em}{0ex}}6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8?\) Yes. | yes | \(1290÷2=645\) |
| \(3\) | \(\text{Is sum of digits divisible by}\phantom{\rule{0.2em}{0ex}}3?\)
|
yes | \(1290÷3=430\) |
| \(5\) | Is last digit \(5\) or \(0?\) Yes. | yes | \(1290÷5=258\) |
| \(10\) | Is last digit \(0?\) Yes. | yes | \(1290÷10=129\) |
Thus, \(1,290\) is divisible by \(2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.\)
Example
Determine whether \(5,625\) is divisible by \(2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.\)
Solution
The table applies the divisibility tests to \(5,625\) and tests the results by finding the quotients.| Divisible by…? | Test | Divisible? | Check |
|---|---|---|---|
| \(2\) | Is last digit \(0,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}4,\phantom{\rule{0.2em}{0ex}}6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8?\) No. | no | \(5625÷2=2812.5\) |
| \(3\) | \(\text{Is sum of digits divisible by}\phantom{\rule{0.2em}{0ex}}3?\)
|
yes | \(5625÷3=1875\) |
| \(5\) | Is last digit is \(5\) or \(0?\) Yes. | yes | \(5625÷5=1125\) |
| \(10\) | Is last digit \(0?\) No. | no | \(5625÷10=562.5\) |
Thus, \(5,625\) is divisible by \(3\) and \(5,\) but not \(2,\) or \(10.\)
Optional Video: Divisibility Rules
This lesson is part of:
The Language of Algebra