You are Here: Home » Math Tricks » multiples of 3 – an answer check and factoring trick

## multiples of 3 – an answer check and factoring trick

So here is a neat math trick to help make some sense of how numbers combine when dealing with 3 and its multiples (3,6,9,12…).  This falls in the “cool, but wierd, but oddly useful” bucket.

Any time you multiply by 3, the digits in the answer will always add up to a multiple of 3.  For example: 8 * 3 = 24, the 2 and 4 add up to 6, which is a multiple of 3 (divide 6 by 3, you get a whole number 2, no remainder, no decimals…).  26 * 3 = 78 – the 7 and 8 sum to 15 – a multiple of 3! Try a few and prove this to yourself.

This also means that any multiple of 3 will, when multiplied by another number, result in a number whose digits also add up to a multiple of 3. For example, 21 * 4 = 84 (notice 21 is a multiple of 3, 7 * 3 = 21) and the 8 and 4 in the 84 add up to 12, which is a multiple of 3!

#### Factoring Trick

Something else to keep in mind is applying this trick in the opposite direction – if the digits of a number add up to a multiple of 3, then that number can be divided by 3.  Take a large number that you don’t often use, like 4356.  Add up the digits, 4 + 3 + 5 + 6 = 18, a multiple of 3!  So, we know that this number is can be divided by 3 with a whole number as a result. 4356 ÷ 3 = 1452  Now, look at the answer, can 1452 be divided by 3?  1 + 4 + 5 + 2 = 12  Yes!  So, 1452 ÷ 3 =  484

This answer doesn’t pass the test (4 + 8 + 4 = 16), so we can’t divide it evenly by 3 🙁 but let’s look at what we know so far… We have divided by 3 twice, so we know that 3 * 3 * 484 = 4356  We could now factor 484, which is a bit easier to handle than the original 4356. This trick is extremely helpful in factoring, and it also can help find an answer to a division problem when you don’t know where to start.