The Least Common Multiple (L.C.M) of two numbers is 144 and their Greatest Common Divisor (G.C.D) is 2. What are the numbers, given their sum is 34.

The Least Common Multiple (L.C.M) of two numbers is 144 and their Greatest Common Divisor (G.C.D) is 2. What are the numbers, given their sum is 34. Correct Answer 18, 16

Given:

L.C.M = 144

G.C.D or H.C.F = 2

Sum of the two number = 34

Formula used:

Product of two numbers = L.C.M × H.C.F

(x + y)² = x2 + y2 + 2xy

(x – y)2 = x2 + y2 – 2xy

Calculation:

Let the two number be x and y

Now, Product of two numbers = L.C.M × H.C.F

⇒ xy = 144 × 2 = 288      ----(1)

And, x + y = 34      ----(2)

(x + y)² = 34²

⇒ x² + y² + 2xy = 1156

⇒ x2 + y2 + 2 × 288 = 1156

⇒ x2 + y2 = 1156 – 576 = 580

⇒ x2 + y2 = 580      ----(3)

Now, (x – y)² = x2 + y2 – 2xy

⇒ (x – y)2 = 580 – 2 × 288 = 4

⇒ (x – y) = ±2      ----(4)

From eq (2) and eq (4), Taking positive value

x + y + x – y = 34 + 2

⇒ 2x = 36

x = 18 and y = 16

When we take negative value then, 

x= 16 and y = 18

∴ The two numbers are 18 and 16