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I know:

1. If x is divided by 30 the remainder is 4.
2. If y is divided by 30 the remainder is 18.

Now the question is how do you show that x*y divided by 30 will give the remainder 12.

I know that I can brute force it, like so:

34/30 --> remainder will be 4, so x=34
48/30 --> remainder will be 18, so y=48

This gives:

(34*48)/30 --> remainder will become 12 which was to be proven.

However I'm wondering if there is a more "elegant" way of showing this, in a more logical way, rather than just guessing some numbers and putting them together?

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    $\begingroup$ the arithmetic would have been simpler if you had chosen $4$ and $18$ rather than $34$ and $48$ $\endgroup$ Nov 5, 2019 at 19:00
  • $\begingroup$ You have to understand that you did not brute-force anything, you did one example. $\endgroup$ Nov 5, 2019 at 19:02
  • $\begingroup$ Yeah, I see that now. But how do I show all examples? @ArnaudMortier $\endgroup$
    – Blue shirt
    Nov 5, 2019 at 19:06
  • $\begingroup$ @J.W.Tanner has given you a perfect answer. $\endgroup$ Nov 5, 2019 at 19:07

1 Answer 1

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$(30m+4)(30n+18)=900mn+540m+120n+72=30(30mn+18m+4n+2)+12$

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  • $\begingroup$ $4\times18=72= 2\times30+12$ $\endgroup$ Nov 5, 2019 at 19:06

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