Track / Stage 3 / 44 of 260 #44 of 1964

Problem 44

AMC 10/12, early questions

Number theory Difficulty 3.2 Find the answer

Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$ . Then
$\mathrm{(A) \ }100<n\leq110$ $\mathrm{(B) \ }110<n\leq120$ $\mathrm{(C) \ } 120<n\leq130$ $\mathrm{(D) \ }130<n\leq140$ $\mathrm{(E) \ } 140<n\leq150$

Your answer

Multiple choice: answer with the letter of the option you want.

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

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.