4, find a number x such that x = 1 mod 4, x 2 mod 7, and x 5 mod 9.

4, 7 and 9 are mutually coprime, so you can use the Chinese remainder theorem.

Start with

Taken mod 4, the last two terms vanish and we're left with

We have , so we can multiply the first term by 3 to guarantee that we end up with 1 mod 4.

Taken mod 7, the first and last terms vanish and we're left with

which is what we want, so no adjustments needed here.

Taken mod 9, the first two terms vanish and we're left with

so we don't need to make any adjustments here, and we end up with .

By the Chinese remainder theorem, we find that any such that

is a solution to this system, i.e. for any integer , the smallest and positive of which is 149.

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step-by-step explanation:

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