1960 AMC 12 Problem 33

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

Every

n

from

2

through

59

has a prime divisor at most

59

That same prime divisor divides both

P

and

n

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.

Problem 33 in other years