Let’s see how we use the chart to deal with this question:
How many positive factors does 100 have?
Set up the chart:
100
| Left Column | Right Column |
|---|---|
| 1 | 100 |
| 2 | 50 |
| 4 | 25 |
| 5 | 20 |
| 10 | 10 |
Remember the rule – once the factors repeat themselves, (i.e. 10 and 10), stop.
Since 10 cannot be counted twice as a factor, you’re left with 9 factors: 1, 2, 4, 5, 10, 20, 25, 50, 100.
What we have here is a special case of finding the number of factors for a perfect square – an integer whose square root is also an integer. The rule here is that every factor can only be counted once.
The quick method for counting the number of factors of a perfect square remains the same, with a slight twist:
1) Test divisibility of the integer by all integers from 1 to the square root of the integer. Use the rules of divisibility to ask “Is this Integer divisible by 1? by 2? by 3?” etc. If you don’t know the rules of divisibility yet, don’t worry about it – we’ll see those later on.
2) Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table.
3) Count the number of small factors and multiply by 2.
4) Subtract 1 from the total – the square root cannot be counted twice. The result is the number of factors.
Another example:
How many positive factors does 144 have?
Since the square root of 144 is 12, we need to check divisibility up to 12. Set up the table:
144
| Left Column | Right Column |
|---|---|
| 1 | .. |
| 2 | .. |
| 3 | .. |
| 4 | .. |
| 6 | .. |
| 8 | .. |
| 9 | .. |
| 12 | … |
8 small factors × 2 = 16. Since 144 is a perfect square, subtract 1 to get 15 factors.
To sum up:
The quick method for counting the number of positive factors of a perfect square remains the same, with a slight twist:
1) Test divisibility of the integer by all integers from 1 to the square root of the integer. Use the rules of divisibility to ask “Is this Integer divisible by 1? by 2? by 3?” etc.
2) Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table.
3) Count the number of small factors and multiply by 2.
4) Subtract 1 from the total – the square root cannot be counted twice. The result is the number of positive factors.