Which of the following is/are prime numbers?(A) 241(B) 337(C) 391(D)…

The correct answer is Option (4) → (D), (A) and (B) only

A number is prime if no prime up to its square root divides it, because a composite number always has a factor no larger than its square root. So each number is divided by the primes up to its square root.

(A) 241 Correct.

• \(15^2=225\) and \(16^2=256\), so the primes 2, 3, 5, 7, 11 and 13 are tested

• None divides 241 (\(7\times34=238\), \(11\times21=231\), \(13\times18=234\)), so 241 is prime

(B) 337 Correct.

• \(18^2=324\) and \(19^2=361\), so the primes up to 17 are tested

• None divides 337 (\(7\times48=336\), \(11\times30=330\), \(13\times25=325\), \(17\times19=323\)), so 337 is prime

(C) 391 Incorrect.

• \(391=17\times23\), so it has factors other than 1 and itself and is not prime

(D) 571 Correct.

• \(23^2=529\) and \(24^2=576\), so the primes up to 23 are tested

• None divides 571 (\(7\times81=567\), \(11\times51=561\), \(13\times43=559\), \(17\times33=561\), \(19\times30=570\), \(23\times24=552\)), so 571 is prime

Tip: A number just below a perfect square can often be split as a difference of two squares: \(391=400-9=20^2-3^2=(20-3)(20+3)=17\times23\).