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Abstract Algebra

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Range of Solutions for Chinese Remainder Theorem

ABSALG-SGQNAI

Let $p,q,r$ be distinct prime numbers. In which of the following ranges must the smallest positive solution $n$ to the congruences:

$$n \equiv 2 \pmod{p}, n \equiv 3 \pmod{q}, n \equiv 5 \pmod{r}$$

...lie?

A

$0 \leq n \leq pqr-1$

B

$pqr \leq n \leq p^2q^2r^2-1$

C

$0 \leq n \leq 29$

D

There is no positive solution to these congruences