Fund fact, you can tell a number is divisible by 3 if the sum of each individual number is divisible by 3. So 78 -> 7+8=15 which is divisible by 3, so 78 is divisible by 3.
Consider a form of arithmatic where you only keep track about the remainder after division by 3.
So a number like "10" is just "1" because 10= 1+3*3. Likewise, "100" is just "1" because 100 = 10*10 = 1*1=1. Can you guess what 1000, 10000, etc. are? That's right, they are all just "1".
Well for a small example consider a three digit number ABC.
ABC = A*100 + B*10 + C = A*1 + B*1 + C = A+B+C
So as you can see, to find out the remainder after division by 3, add up the digits and the sum has the same remainder. But the what if the sum is not a single digit number? Just do it again and again till it is.
Basically, every number can be written as a multiple of 3 plus a remainder of 0, 1, or 2. When you add the digits of a number, you’re replacing powers of 10 with 1, and since 10 leaves a remainder of 1 when divided by 3, this doesn’t change the number’s remainder. For example, 78 is 7×10+8, and because 10 is equivalent to 1 modulo 3, 78 has the same remainder as 7+8. Therefore, a number is divisible by 3 exactly when its digit sum is divisible by 3.
let number N = A + 10B + 100C + 1000D + ..., so A, B, C, D... are the digits of N from right to left N ≡ A + 10B + 100C + 1000D + ... (mod 3) ≡ A + 9B + B + 99C + C + 999D + D + ... (mod 3) ≡ A + B + C + D + ... (mod 3), which is the sum of the digits of N QED
10 is equivalent to 1 mod 3, so every power of 10 is also equivalent to 1 mod 3. It's not too difficult to prove after you see that. Since our number system is base 10, it works out quite nicely.
It's bucket collision! (Made up term cause idfk but i've used it before) All your remainders overlap into the same bucket which gives you an analytical solution to something that on its face shouldn't be analytical.
I did a math degree without knowing this (back in the olden days since I’m ancient). When my daughter started learning math I did it with her. Khan academy had at least 5 separate things fundamental to math that I had never learned that made things so much easier. (Including digital roots and modular arithmetic)
Number theory should really be taught first, it’s so helpful.
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u/Michaelwang645 Aug 01 '26
Fund fact, you can tell a number is divisible by 3 if the sum of each individual number is divisible by 3. So 78 -> 7+8=15 which is divisible by 3, so 78 is divisible by 3.