Find the in-congruent solutions to

Working backwards I obtain:

Is this correct and if so, is there an easier way to do this?

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- Jul 4th 2010, 10:54 AMdwsmithCongruence
Find the in-congruent solutions to

Working backwards I obtain:

Is this correct and if so, is there an easier way to do this? - Jul 4th 2010, 01:31 PMHallsofIvy
It is easy to see that it works: 49*10= 490 and 490/36= 13 with remainder 22.

As for an "easier way", what you did looks pretty easy to me! Certainly better than multiplying 48 by 1, 2, 3, etc. until you find the right one. - Jul 4th 2010, 02:00 PMdwsmith
- Jul 4th 2010, 02:19 PMAlso sprach Zarathustra
Maybe...

49x=22(mod 36)

==>

49x+14=36=0 (mod36)

==>

7(7x+2)=0(mod36)

now it's clear that x=10 - Jul 4th 2010, 03:20 PMmelese
- Jul 4th 2010, 03:22 PMAlso sprach Zarathustra
This is "my solution" :)

- Jul 4th 2010, 03:25 PMmelese
- Jul 4th 2010, 06:19 PMSoroban
Hello, dwsmith!

Here is a very primitive solution . . .

Quote:

Solve:. .

The congruence reduces to: .

This means: .

.[1]

Since is an integer, must be divisible by 13.

We have: .

.[2]

Since is an integer, must be divisible by 10.

The first time this happens is when

Substitute into [2]: .

Substitute into [1]: .

Therefore: .

- Jul 4th 2010, 08:43 PMsimplependulum
Or how about this ?

I don't consider the convergence of the series because we find that