Now let p and q be distinct (i.e., different)
primes and let c, d, m and n be integers
such that c2 = m (mod p) and d2 = m (mod q).
Assume x=n is a simultaneous solution to
x = c (mod p)
x = d (mod q)
Show that n is a solution to x2 = m (mod pq).
[Hint: Use the given congruences to
show p and q both divide n2 - m. Then justify why that means
that pq divides n2 - m.]
For these last two problems, use the ideas of the preceding two
problems to find all least positive residues which are
solutions, or to show that there are no solutions.