# Factorising Tricks

### Table of Factorising Tricks

How do we know if a number is divisible by a particular smaller number? It would certainly make factorisation easier. The most straight forward way is recognising the number from our times tables. There are also other ways we can break numbers down to help spot particular factors.

With tricks for finding factors up to 31, all numbers up to more than 1,000 can be factorised.

### Examples

The trick for figuring out if a number is divisible by 3 is quite handy, and is the basis for figuring out if numbers are divisible by 6 and 9 as well. If we have a big, daunting number, just add the digits together. If we end up with more than one digit in the sum and can't easily recognise it as a multiple of three we can simply do it again. (In other words, the rule can be applied recursively.)

123,456 divisible by 3?
1 + 2 + 3 + 4 + 5 + 6 = 21
2 + 1 = 3

And since 3 is a multiple of 3 we know that 21 is a multiple of 3, which means 123,456 is a multiple of 3. We also know that 123,456 is a multiple of 6 since it's also even. We also know that 654,321 must be divisible by three (but is not a multiple of 6).

Let's try a bigger number. How can we tell at a glance if 123,456,789 is a multiple of 9? If the digits add to a multiple of 9 we will know it's divisible by 9.

123,456,789 divisible by 9?
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
4 + 5 = 9

So yes, 123,456,789 is divisible by 9.

Another way of doing it is that we can simply ignore groups of digits if they add to 9. This is because adding a 9 will not change whether the sum is a multiple of 9 – it'll just be a different multiple of 9.

123,456,789 divisible by 9?
The 9 on the end is, so
Yes if 12,345,678 is divisible by 9
1 + 8 = 9, so
Yes if 234,567 is divisible by 9
2 + 7 = 9, so
Yes if 3,456 is divisible by 9
3 + 6 = 9, so
Yes if 45 is divisible by 9
4 + 5 = 9

Let's use these tricks to factorise 6,831 – the number of lithium ion cells in the battery of a Tesla electric car. The first thing I notice is that it's not even, but 8+1 makes 9, which is a multiple of 3, and since 6 and 3 are already multiples of 3, the whole 6,831 must be as well. The next thing I notice is that it's divisible by 9, since 6+3=9 and 8+1=9. So 3 and 3 are factors of 6,831. No calculators, now.

 6,831 = 3 x 2,277 = 3 x 3 x 759 (still divisible by 3) = 3 x 3 x 3 x 253 (3 digits with that pattern, must be divisible by 11) = 3 x 3 x 3 x 11 x 23

Other Tesla batteries include their "85" kWh pack (16 modules of 444 cells; 7,104 total cells) and their "60" kWh pack (14 modules of 384 cells; 5,376 total cells).

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