Originally Posted by

**jsndacruz** I feel really silly not being able to figure this out -- the problems come from a proof an example of the Chinese Remainder Theorem. I have tried multiplying and adding and subtracting and such on scrap paper but the answers simply don't make sense to me. The answer won't help me so much as a description of the method, so really any advice is appreciated!!. Thanks in advance! Oh, and take == to meant congruent to...

1) A claim (book doesn't bother to explain why) states that all sol'ns of 7x == 1 (mod 31) satisfy x == 9 (mod 31)

2) Another says 5t + 1 == 2 (mod 6) can be solved for t to yield t == 5(mod 6)

3) The last one says 30u + 26 == 3(mod 7) can be solved for u to yield u == 6 (mod 7)

The first one -- WHY?! If 31 divide 7x-1, does that mean 31 divides x-9?!

The second one -- I don't even know what to do. Subtracting 1 from each side gets 5t == 1(mod 6). Using Euclidean Algo (or common sense), gcd(5,6)=1, and 1 = 1.6 - 1.5 so t=1 works. So 1 == 5 (mod 6). Does this mean that t == 5 (mod 6)?

The last one I start off by subtracting 26 to get 30u == -23 (mod 7). I believe you can subtract off multiples of 7 from 20, to get 2u == -23 (mod 7) . [Is the same allowed on the right?]. Since gcd(2,7)=1 which obv divides 23, We can use 1 = 1.7 - 3.2 --> -23 = -23.7 + 69.2 so u=69. So 2(69) == -23(mod 7). But then what is u congruent to mod 7?

AH!