The Least Common Multiple and Highest common Factor of two numbers are 30 and 5 respectively. If their sum is 25, then what will be the difference of these two numbers?

Question

Question

This question was previously asked in

SSC GD Previous Paper 7 (Held On: 13 Feb 2019 Shift 1)

Answer 1

Concept:

The largest common factor of two or more numbers is called the highest common factor.

The smallest positive number that is a multiple of two or more numbers is called Least common multiple

Formula Used:

LCM × HCF = Product of two number

Calculation:

Let the number be a and b

LCM × HCF = a × b

⇒ a × b = 150 ---- (1)

Given than,

⇒ a + b = 25 ----(2)

Taking square on both sides,

⇒ (a + b)² = 25²

⇒ a² + b² + 2ab = 625

⇒ a2 + b2 + 2(150) = 625

⇒ a2 + b2 = 625 – 300 = 325 ----(3)

(a – b)² = a² + b² – 2ab

⇒ (a – b)2 = 325 – 2(150)

⇒ (a – b)2 = 325 – 300 = 25

⇒ a – b = 5 ----(4)

Adding equation (2) and (4),we get,

⇒ a + b + a – b = 25 + 5

⇒ 2a = 30

⇒ a = 30/2 = 15

Substituting a = 15, in equation (4)

⇒ a – b = 5,

⇒ 15 – b = 5

⇒ b = 15 – 5 = 10

∴ required difference of these two numbers = 15 – 10 = 5