Why is the sum of the digits of 654321÷3 congruent to 654321÷3 (mod 3)?

divinedolphin

debbie

@divinedolphin I agree, the way this is written, it looks a little bit confusing! I have a feeling that you understand the basic principle of what Prof. Loh was talking about, but just aren't very used to the notation. For example, if I write

$$ 25 + 36 + 37 \text{ ( mod 3 ) }, $$

it doesn't mean

$$ 98. $$

Rather, it means the remainder of $ 25 + 36 + 37 $ after dividing by $3.$ There's a sneaky clue here that gives this away: the not-equals sign!

Do you see the weird equal sign? It's not a $ =,$ it's a $ \equiv.$ This tells you that we are only caring about the remainder of a numerical expression after dividing that expression by a certain number, which we call "the modulus," which in this case is $3.$

So,

$$ 25 + 36 + 37 \text{ ( mod 3 ) } \equiv 2.$$

Now, when Prof. Loh says "the sum of the digits of $654321 \div 3 \equiv 654321 \div 3 \text{ ( mod 3 )"},$ he means
that you first find the sum of the digits of $654321,$ and then divide that sum by $3,$ not that you first divide $654321$ by $3$ and then take the sum of the digits.

The next key detail here is that Prof. Loh decides to find the sum of the digits of $654321$ and find its remainder after dividing by $9$ first. Why did he do that? Well, if he tried dividing $654321$ first by $3,$ you would find that

$$ 654321 = \text{ something } \times 3 + \text{ remainder},$$

but you wouldn't immediately know anything about the "something," like, for example, if it is divisible by $3.$ So the cool math insight here is that if you know that

$$ 654321 = \text{ something } \times 9 + \text{remainder},$$

at least we can just focus all of our attention on that remainder, because the first term, after dividing by $3,$ is automatically divisible by $3.$ We only need to worry about the remainder and what it will give after dividing by $3.$ We only need to look at the remainder of the remainder, instead of the whole huge gigantic troublesome number.

divinedolphin

Thank you so much.

debbie

@divinedolphin You're most welcome!