Question

A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

Answer 1

Let the two-digit number = 10x + y

Case I: Multiplying the sum of the digits by 8 and then subtracting 5 = two-digit number

⇒ 8 × (x + y) – 5 = 10x + y

⇒ 8x + 8y – 5 = 10x + y

⇒ 2x – 7y = –5   .....(i)

Case II: Multiplying the difference of the digits by 16 and then adding 3 = two-digit number

⇒ 16 × (x – y) + 3 = 10x+ y

⇒ 16x – 16y + 3 = 10x + y

⇒ 6x – 17y = –3  ......(ii)

Now, multiplying equation (i) by 3 and then subtracting from equation (ii), we get

(6x – 17y) – (6x – 21y) = – 3 – (–15)

⇒ 4y = 12

⇒ y = 3

Now, put the value of y in equation (i), we get

2x – 7 × 3 = –5

⇒ 2x = 21 – 5 = 16

⇒ x = 8

Hence, the required two-digit number

= 10x + y

= 10 × 8 + 3

= 80 + 3

= 83