The GCD of two numbers a and b is 4 and their LCM is 400. The number of pairs of a and b are

Given:

GCD(a, b) = 4

LCM(a, b) = 400

Formula used:

Product of two numbers = GCD × LCM

a × b = GCD(a, b) × LCM(a, b)

Let a = 4m and b = 4n, where GCD(m, n) = 1.

Calculation:

a × b = GCD(a, b) × LCM(a, b)

⇒ 4m × 4n = 4 × 400

⇒ 16mn = 1600

⇒ mn = 100

Find pairs (m, n) such that GCD(m, n) = 1 and mn = 100:

Possible pairs are:

(m, n) = (1, 100), (100, 1), (4, 25), (25, 4).

Each pair gives distinct values of a and b:

(a, b) = (4 × 1, 4 × 100) = (4, 400)

(a, b) = (4 × 100, 4 × 1) = (400, 4)

(a, b) = (4 × 4, 4 × 25) = (16, 100)

(a, b) = (4 × 25, 4 × 4) = (100, 16)

∴ The number of pairs of (a, b) is 2.

The correct answer is option (3).