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# How many positive integers $n$ satisfy $127 \equiv 7 \pmod{n}$? $n=1$ is allowed.

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How many positive integers $n$ satisfy $127 \equiv 7 \pmod{n}$? $n=1$ is allowed.

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Hey RB!

If $$127 \equiv 7 \pmod{n}$$, then n is a divisor of 127 - 7 = 120.

The prime factorization of 120 is $$2^3 \cdot 3 \cdot 5$$

which has $$(3 + 1)(1 + 1)(1 + 1) = 16$$ positive divisors.

Therefore, there are 16 possible values of n.

I hope this helps,

Gavin.

GYanggg  Apr 30, 2018