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# Number Theory

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How many integers n satisfy 0 < n < 60 and 4n=6 (mod 7)

Sep 6, 2021

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There might be an easier way to do this but this is what I have learnt. (I think from Heureka)  Thanks Heureka

How many integers n satisfy 0 < n < 60 and 4n=6 (mod 7)

4n=6  (mod 7)

$$4n=6\mod 7\\ 4n=7k+6\\ 4n=8k+4-k+2\\ n=(2k+1)+\frac{2-k}{4}\\ let \;\;a=\frac{2-k}{4}\\ \text{after rearranging }k=2-4a\\ \text{substituting back in}\\ n=2(2-4a)+1+a\\ n=5-7a\\ \text{n is between 0 and 60}\\ n=5,12,19,26,33,40,47,54$$

So I found 8.

Sep 6, 2021