Proofs and numbers
posted by yin .
Solve the congruence 24x identical to 42 (mod 30) for x. List all the solutions (if there are any).

24x = 42 mod 30
24x = 12 mod 30
(2430)x = 12 mod 30
6x = 12 mod 30
x = 2
so one possible solution is x = 2(24) = 48
adding/subtracting multiples of 30 , our mod, will yield other solutions
making 2(24) + 60 , or 12 , our smallest positive answer
A general solution would be 12 + 30k, where k is an integer
check:
look at the sequence of numbers
... 108  78  48 18 12 42 72 102 ...
IF we take 102 ÷ 30 we get
3 + 12/30
or
2 + 42/30
so clearly we can see that 102 = 12 mod 30 = 42 mod 30