\(n\) is an integer with \(20\lt n\lt 40\). (a) Write down all the values of \(n\) that are prime numbers. [2] (b) Write down the value of \(n\) that is a c...

The condition \(20 \lt n \lt 40\) is strict at both ends, so the possible values run from \(21\) to \(39\).

(a) A prime number has exactly two factors, itself and \(1\). Testing each candidate for divisibility by \(2\), \(3\) and \(5\) removes almost all of them, and it is enough to test primes up to \(6\), since \(6^{2} = 36\) is near the top of the range and \(7^{2} = 49\) is above it.

The primes in the range are \(23,\ 29,\ 31,\ 37\). Three of these correct with no incorrect extras earns [B1], and the complete list earns the second [B1].

Numbers such as \(21 = 3 \times 7\), \(27 = 3 \times 9\), \(33 = 3 \times 11\) and \(39 = 3 \times 13\) are the ones most often included by mistake.

(b) The cube numbers near this range are \(2^{3} = 8\), \(3^{3} = 27\) and \(4^{3} = 64\). Only \(27\) lies between \(20\) and \(40\), so \(n = 27\) [B1].

(c) The square numbers are \(4^{2} = 16\), \(5^{2} = 25\), \(6^{2} = 36\) and \(7^{2} = 49\). Those in range are \(25\) and \(36\) [B1].