UPSC CDS 1 2020 Math Paper

Concept:A number that has exactly two factors (1 and itself) is called a prime number.A composite number has more than two factors.Explanation:First, compute the given expression:$2 \times 3 \times 5 \times 7 \times 11 + 1 = 2310 + 1 = 2311$.Now check if $2311$ is prime.It is not divisible by $2, 3, 5, 7,$ or $11$ (each gives remainder $1$).Test divisibility by primes up to $\sqrt{2311} \approx 48$: $13, 17, 19, 23, 29, 31, 37, 41, 43, 47$.$2311 \div 13 \approx 177.77$ (remainder $10$); $2311 \div 17 \approx 135.94$; $2311 \div 19 \approx 121.63$; $2311 \div 23 \approx 100.48$; $2311 \div 29 \approx 79.69$; $2311 \div 31 \approx 74.55$; $2311 \div 37 \approx 62.46$; $2311 \div 41 \approx 56.37$; $2311 \div 43 \approx 53.74$; $2311 \div 47 \approx 49.17$.None divide exactly, so $2311$ has only two factors: $1$ and $2311$.Thus it is a prime number.Answer:A. a prime number