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

N = p⋅ q⋅ r + q

N = q(p⋅ r + 1)

Since q is a prime number and p⋅r + 1 is an integer greater than 1, N is the product of two numbers greater than 1. This means N is always composite.

Thus, p⋅q⋅r + q is always a composite number.

(i) P.q.r + 1 is a composite

Let's choose three distinct prime numbers, say:

p = 2, q = 3, r = 5

N = p⋅q⋅r + 1

= (2 × 3 × 5) + 1

= 30 + 1

= 31

Since 31 is a prime number, this does not satisfy the composite case.

p = 2, q = 3, r = 7

N = (2 × 3 × 7) + 1 

= 42 + 1

= 43

Since 43 is also prime, this still does not satisfy the composite case.

p = 2, q = 3, r = 11

N = (2 × 3 × 11) + 1

= 66 + 1

= 67

Since 67 is prime, we need a different example.

p = 2, q = 3, r = 13

N = (2 × 3 × 13) + 1

= 78 + 1

= 79

Since 79 is also prime, let's attempt different primes.

p = 2, q = 5, r = 7

N = (2 × 5 × 7) + 1

= 70 + 1

= 71

p = 3, q = 5, r = 7

N = (3 × 5 × 7) + 1

= 105 + 1

= 106

Since 106 is composite (divisible by 2), this is an example of when p⋅q⋅r + 1 is composite.

(ii) p⋅q⋅r + 1 is Prime

(2 × 3 × 5) + 1 = 31

Since 31 is prime, this is an example where p⋅q⋅r + 1 is prime.