One new thing I learned in math this week was divisibility rules with numbers outside of 2, 5, 10, etc. being able to tell just by looking at a large number what its divisible by.

The divisibility rule for 3 is a bit clever, the rule goes that you add up each digit of the number and then check to see if the sum of all the digits is divisible by 3. Here’s an example, if you add all of the digits of 456 together, doing 4+5+6, you would get 15 which 3 goes into 5 times, that sum of the digits being divisible by 3 means that the number itself is also divisible by 3. An example of a number that wouldn’t work would be 626 as adding 6+2+6 would give you 14 which 3 does not go into, meaning it is not divisible by it. The rule for 9 works very similarly but the sum of the digits instead have to be divisible by 9, under these rules 297 would work as a multiple of 9 as 2+9+7 is 18 which 9 goes into twice.

Seeing if a number is divisible by 4 has two steps to it, first, the number has to be even (last digit being 0, 2, 4, 6, or 8) and second, the last two digits have to be able to half twice, using 244 as an example, you’d take the last 2 digits being 44 and half it (divide by 2) twice, the first time halving it gives you 22, and a second time gives you 11, since we were able to half the last two digits twice without ending up with a decimal the number is divisible by 4. An example of this not working would be with 126, take the last two digits being 26 and half it giving you 13, since we can’t half 13 without getting a decimal 126 is not divisible by 4. The rule for 8 works in a similar way but instead of halving the last two digits of the number twice, you halve the last 3 digits 3 times, a number that this works on would be 1368, halving the last three digits gives us 184, halving that gives us 92, and halving that gives us 46. Since we ended up with no decimals throughout that whole process, 1368 is very much divisible by 8.

Rules for numbers like 6 and 12 work by multiplying two divisors that work to get a larger one, for example if a number is divisible by 3 and 2 it will always be divisible by 6 because 3 x 2 = 6, same goes with 12, if a number is divisible by 3 and 4 it will always be divisible by 12 because 3 x 4 = 12. This can work with any two divisors, not just the ones I mention.

Learning these divisibility rules help a lot when getting the prime factorization of a number as you can immediately tell what to try dividing by to break it down, overall these were some very helpful and surprisingly fun things that I’ve learned this week in math class.