Question

Let p, q and r be three distinct prime numbers.

Check whether p × q × r + q is a composite number or not.

Further, give an example for 3 distinct primes p, q, r such that

1. p × q × r + 1 is a composite number.
2. p × q × r + 1 is a prime number.

Answer 1

Let’s factor the expression p × q × r + q:

p × q × r + q = q(p × r + 1)

Since q is a prime number, it’s greater than 1. Also, p × r is a product of two primes, so p × r > 1, which means (p × r + 1) > 2.

Therefore, q and (p × r + 1) are both greater than 1. This means p × q × r + q is always composite for any distinct primes p, q and r.

i. Let p = 3, q = 5, r = 7

p × q × r + 1

= 3 × 5 × 7 + 1

= 105 + 1

= 106

Since 106 = 2 × 53, it is a composite number.

ii. Let p = 2, q = 3, r = 5

p × q × r + 1

= 2 × 3 × 5 + 1

= 30 + 1

= 31

∴ 31 is a prime number.