You’ve already seen the wonderful factor table – a simple technique for finding all the factors of an integer.
Let’s see how we use the chart to deal with this question:
How many positive factors does 140 have?
The question asks about factors, so the factor table is appropriate.
Set up the table:
140
| Left Column | Right Column |
|---|---|
| 1 | 140 |
| 2 | 70 |
| 4 | ….. |
And let’s pause for a second.
Because 140/4 is the first calculation that requires a bit of work. Which brings us to the question: is this work necessary?
After all, the question asked how many numbers are factors of 140, and not what are the factors. Calculating that 140/4 = 35 is unnecessary – in order to answer the question, all we really need do is count factors.
Think about it: for each of the small numbers on the left column on the table, there will be a bigger counterpart on the right. In fact, all you need to do is find the number of small numbers that 140 is divisible by, and multiply that number by 2 – a bigger counterpart for every small number.
Instead of asking yourself “what is the result of dividing 140 by 4?”, just ask yourself “is 140 divisible by 4?”
Moreover, we already know the rules of divisibility.
140 is divisible by 4 because the number formed by the last two digits (40) is divisible by 4.
The chart now looks like this:
140
| Left Column | Right Column |
|---|---|
| 1 | .. |
| 2 | .. |
| 4 | .. |
| 5 | .. |
| 7 | .. |
| 10 | .. |
And the number of factors of 140 can be calculated by counting the number of factors on the left and multiplying by 2. Six small factors*2 = 12 factors. So, 12 is the answer.
Sure thing. But you don’t really need them.
The chart now looks like this:
140
| Left Column | Right Column |
|---|---|
| 1 | 140 |
| 2 | 70 |
| 4 | 35 |
| 5 | 28 |
| 7 | 20 |
| 10 | 14 |
Satisfied? Did you really need to know that 140/5 = 28 in order to answer the question? Of course not. That’s a needless calculation if ever there was one.
Without listing all those numbers on the right, you may want to know when to stop with the left column. That’s a very good question. The method for the factor chart was to stop when the factors repeat themselves. But if you don’t calculate all the factors in the right side of the table, you don’t know when the factors start repeating themselves.
Here’s the last bit of wisdom regarding the factor chart: Stop when you reach the square root of the original integer. Why this is so is less important – trust us, it’s enough.
The square root of 140 is ?140 = 11.8. Therefore, check until 11, then stop.
Remember, that you have a calculator with a square root function so you can use that to find the square root of 140.
To sum up:
For questions with large numbers that ask for the number of positive factors of an Integer:
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. That’s the number of factors.