# Proofs and numbers

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Solve the congruence 24x identical to 42 (mod 30) for x. List all the solutions (if there are any).

• Proofs and numbers -

24x = 42 mod 30
24x = 12 mod 30

(24-30)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

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