1984 AHSME Problems/Problem 3

Problem

Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then

$\mathrm{(A) \ }100<n\leq110 \qquad \mathrm{(B) \ }110<n\leq120 \qquad \mathrm{(C) \ } 120<n\leq130 \qquad \mathrm{(D) \ }130<n\leq140 \qquad \mathrm{(E) \ } 140<n\leq150$

Solution

Since the number isn't prime, it is a product of two primes. If the least integer were a product of more than two primes, then one prime could be removed without making the number prime or introducing any prime factors less than $10$. These prime factors must be greater than $10$, so the least prime factor is $11$. Therefore, the least integer is $11^2=121$, which is in $\boxed{\text{C}}$.

See Also

1984 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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