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When a two digit number ab, where b > a, is multiplied by the sum of its digits, the number 52 is obtained. When the number obtained by interchanging the digits of the given number is multiplied by the sum of its digits, the number is 124. The value of (14a - 3b) is:

When a two digit number ab, where b > a, is multiplied by the sum of its digits, the number 52 is obtained. When the number obtained by interchanging the digits of the given number is multiplied by the sum of its digits, the number is 124. The value of (14a - 3b) is: Correct Answer 5

Given:

A two digit number ab, where b > a, is multiplied by the sum of its digits, the number 52 is obtained.

The number obtained by interchanging the digits of the given number is multiplied by the sum of its digits, the number is 124.

Concept used:

Number = (10a + b)

Where, b > a

Calculation:

According to the question:

(10a + b)(a + b) = 52

⇒ 10a2 + b2 + 11ab = 52     ----(1)

(10b + a)(a + b) = 124

⇒ 10b2 + a2 + 11ab = 124     ----(2)

Subtract from equation(1) to equation(2):

(10a2 + b2 + 11ab) - (10b2 + a2 + 11ab) = 52 - 124

⇒ 10a2 + b2 + 11ab - 10b2 - a2 - 11ab = -72

⇒ 9a2 - 9b2 = -72

⇒ b2 - a2 = 8     ----(3)

Now,

/= 124/52

⇒ 130b + 13a = 310a + 31b

⇒ 99b = 297a

⇒ b = 3a

From equation(3);

b2 - a2 = 8 

⇒ (3a)2 - a2 = 8

⇒ 9a2 - a2 = 8

⇒ 8a2 = 8

⇒ a2 = 1

⇒ a = 1

So, b = 3a = 3 × 1 = 3

(14a - 3b) = 14 × 1 - 3 × 3 = 14 - 9 = 5

∴ The value of (14a - 3b) is 15.

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