The prime numbers 23 and 29 are consecutive prime numbers since 29 is the smallest prime number that is greater than the prime number 23 . How many positive integers less than 900 can be written as a product of two or more consecutive prime numbers?
(A) 14
(B) 13
(C) 11
(D) 12
(E) 15
We begin by listing the prime numbers up to and including 31 .

We choose to end the list at 31 since $30 \times 30=900$ and so $31 \times 37$ is greater than 900 .

This list of prime numbers is

$$
2,3,5,7,11,13,17,19,23,29,31
$$

Considering consecutive pairs of prime numbers from the list above, we get 10 positive integers less than 900 that can be written as a product of two consecutive prime numbers. These are

$$
\begin{array}{ccccc}
2 \times 3=6 & 3 \times 5=15 & 5 \times 7=35 & 7 \times 11=77 \quad 11 \times 13=143 \\
13 \times 17=221 & 17 \times 19=323 & 19 \times 23=437 \quad 23 \times 29=667 \quad 29 \times 31=899
\end{array}
$$

There are 3 positive integers less than 900 that can be written as a product of three consecutive prime numbers. These are

$$
2 \times 3 \times 5=30 \quad 3 \times 5 \times 7=105 \quad 5 \times 7 \times 11=385
$$

Since each positive integer greater than 1 is either a prime number or can be written as a unique product of prime numbers, it must be that these 3 numbers, 30, 105, 385, are different than those obtained from a product of two consecutive prime numbers. (Alternatively, we could check that 30, 105 and 385 do not appear in the previous list.)

Further, we note that the next smallest positive integer that can be written as a product of three consecutive prime numbers is $7 \times 11 \times 13=1001$ which is greater than 900 .

Exactly one positive integer less than 900 can be written as a product of four consecutive prime numbers. This number is

$$
2 \times 3 \times 5 \times 7=210
$$

The next smallest positive integer that can be written as a product of four consecutive prime numbers is $3 \times 5 \times 7 \times 11=1155$ which is greater than 900 .

There are no positive integers that can be written as a product of five or more consecutive prime numbers since $2 \times 3 \times 5 \times 7 \times 11=2310$ which is greater than 900 .

In total, the number of positive integers less than 900 that can be written as a product of two or more consecutive prime numbers is $10+3+1=14$.