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QUESTION
Is 157 prime? What about 221?
ANSWER
To test whether $n$ is prime, we need only check whether $n$ is
divisible by any prime $\leq\sqrt{n}$.
Now $12^2<157<13^2$ and $14^2<221<15^2$, so we must test 157 for
divisibility by the primes 2,3,5,7,11 and 221 for divisibility by
the primes 2,3,5,7,11,13.
Quick checking methods eliminate 2,3,5 and 11 in both cases, and a
check by hand establishes that 7 divides neither number.
This shows that 157 is prime, but on checking 221 for divisibility
by 13, the factorisation 221=13.17 is found. Thus 221 is not
prime.
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