I need to find minimal x∈N that solves the linear congruences:
6x≡2(mod4)
3x≡6(mod9)
x≡15(mod17)
I divided the first congruence by 2 and the second congruence by 3, then used the Chinese remainder theorem.
I got x≡393(mod2⋅3⋅17)
I checked that this solution actually solves the 3 equations.
But is it the smallest one? If so, then why ?
Thanks.
Answer
Your congruences may be reduced to
x≡1(mod2)
x≡2(mod3)
x≡15(mod17)
We have mod17, x≡15≡32≡49≡66≡83.
And we note that 83 satisfies the other two congruences.
The general solution is then x≡83(mod102)
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