Question

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $\mathrm{2.3.5...61}$ of all prime numbers (a prime number is a number divisible only by $1$ and itself) less than or equal to $61$, and $n$ takes, successively, the value $2,3,4,...,59$. Let $N$ be the number of primes appearing in this sequence. Then $N$ is

• A. $0$
• B. $16$
• C. $17$
• D. $57$
• E. $58$

Answer 1

Answer: (A)

$0$

All those numbers $P+n$ of the sequence $P+2,P+3,...,P+59$ for which $n$ is prime are divisible by $n$ because $n$ already occurs as a factor in $P$. In the remaining members, $n$ is composite and hence can be factored into primes that are smaller than $59$. Hence all terms are divisible by $n$
$\therefore$ all members of the sequence are composite.