I need to find minimal x∈N that solves the linear congruences:
6x \equiv 2 \pmod {\!4}
3x \equiv 6 \pmod {\!9}
x \equiv 15 \pmod {\! 17}
I divided the first congruence by 2 and the second congruence by 3, then used the Chinese remainder theorem.
I got x\equiv 393 \pmod {2\cdot 3\cdot 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\equiv 1\pmod{2}
x\equiv 2 \pmod{3}
x\equiv 15 \pmod{17}
We have \mod {17}, x\equiv 15\equiv 32\equiv 49 \equiv 66 \equiv 83.
And we note that 83 satisfies the other two congruences.
The general solution is then x\equiv 83 \pmod{102}
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