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Official Paper 24: Held on 13th Nov 2020 Shift 2

Option 4 : 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} = 34^{2}

⇒ x^{2} + y^{2} + 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**