Question

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2\cdot3\cdot5\cdots61$ of all prime numbers less than or equal to $61,$ and $n$ takes, successively, the values $2,$ $3,$ $4,$ $\ldots,$ $59.$ Let $N$ be the number of primes appearing in this sequence. Then $N$ is:

$0$

$16$

$17$

$57$

$58$

Answer: A

Concepts:prime divisibility prime factorization

Difficulty rating: 1500

Small Hint:

Every $n$ from $2$ through $59$ has a prime divisor at most $59$

Big Hint:

That same prime divisor divides both $P$ and $n$

Solution:

For each $n$ in the given range, choose a prime divisor $p$ of $n.$ Since $p\le59,$ it is one of the factors of $P.$ Therefore $p$ divides $P+n.$ Moreover $P+n>p,$ so $P+n$ is composite. No term is prime, and $N=0.$

Thus, the correct answer is A.

1960 AMC 12 Problem 33

Answer: A

Difficulty rating: 1500

Small Hint:

Big Hint:

Solution:

Problem 33 in Other Years