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 $\boxed{\text{C}}$ .

1984 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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1984 AHSME Problems/Problem 3

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