Proving a number is divisible by 48

[matt grime]

You're missing out 6m+4, but the point is if you were to put any of those numbers into the formula k^3+5k you get a number divisible by 6. He's just doing the "unclever" way of solving the problem, that is to say he's doing modulo arithmetic without telling you he's doing modulo arithmetic.

Every given any number n and another number p we can express n=pq+r where q and r are integers 0<=r<p and q is then uniquely determined. That is just kindergarten mathematics (which is far more mathematically mature than knowing how to write out decimals if you ask me) where r is the remainder after dividing by p.

All he is now claiming is that whatever the remainder is after dividing by 6 (ie what class it is in modulo r) we get something divisible by 6, eg

if k=6m+1

k^3+5k= (6m+1)^3+5(6m+1) = (6m^3) +3(6m)^2+4(6m)+1+5(6m)+5

obvisouly anyting with a 6m in it is divisible by 6 so that just leaves 1+5 part, which is also divisible by 6.

Hopefully you see that if I expand (6m+r)^3 i get something divisible by 6 plus r^3, and the 5k=30m+5r, so I just need to show that r^3+5r is divisible by 6 for r from 0 to 5.

 

[johnnyICON]

Oooh.. so those forms, 6m, 6m+1, etc., are all possible ways of expressing any number in terms of 6?
That is q=6, the uniquely determined integer? But that doesn't mean that n is divisible by 6?

Oh I get it... so for any integer k, k can be expressed using one of the six forms.
(Now help me with my understanding of why we do this...) We use q=6 so that we may be able to find 6 as a common term... I'm afraid to say it but I think I have the idea...

 

[shmoe]

Oooh.. so those forms, 6m, 6m+1, etc., are all possible ways of expressing any number in terms of 6?
That is q=6, the uniquely determined integer? But that doesn't mean that n is divisible by 6?

In matt's notation p=6, you're dividing your number k by 6, so it's of the form k=6m+r, where r=0, 1, 2, 3, 4, or 5 (you're using m for his q). All we're going to care about for this problem is the r, the m won't matter. k is divisible by 6 exactly when r=0.

Note if k=6m+r then k=r mod 6.

Oh I get it... so for any integer k, k can be expressed using one of the six forms.
(Now help me with my understanding of why we do this...)

We are interested in k^3+5k mod 6, specifically we'd like to show it's zero. Since we are only interested mod 6, we can replace k by r and get k^3+5k=r^3+5r mod 6. In otherwords, the remainder you get when you divide k^3+5k by 6 is dependant only upon the remainder you get when you divide k by 6. e.g. 4^3+5(4)=10^3+5(10) mod 6. So writing k in one of these 6 forms allows us to ignore the unimportant (for this problem) m part and focus on the r. Since we have only 6 options for r, we can try them all and see that in each case we get 0 mod 6.

 

[TenaliRaman]

He's just doing the "unclever" way of solving the problem, that is to say he's doing modulo arithmetic without telling you he's doing modulo arithmetic.

It seems like a good way to introduce modular arithmetic. This comes as a personal experience. Once i was explaining to a friend, why m(m+1)...(m+n) is divisible by n(in our discrete maths class). I started with "consider their remainders modulo n, one of them must be zero, so we are done!". His reaction **blank**. Ok, so i started, see i can express any number as nk,nk+1,nk+2,...,nk+(n-1) and his reaction "what? how can u do that?". I gave him some examples and he looked at me as if i did some magic! So now i had to show why this is true which took me to the equation,
Dividend = Divisor*Quotient + Remainder
Now he was well aware of this, then i made a connection of this equation with the above proposed "i can express any number as...". Then i had to get back to the original question. To this day, i still hope he understood what i said that day.

-- AI