A three digit number is equal to 17 times the sum of its digits. If 198 is added to the number, the digits are interchanged. The addition of first and third digit is 1 less than middle digit. Find the number.

#### Solution

Let the three-digit number be xyz.

Its numerical value = 100x + 10y + z

According to first information provided in the question,

100x + 10y + z = 17(x + y + z)

∴ 100x + 10y + z = 17x + 17y + 17z

∴ 83x – 7y – 16z = 0 ....(1)

Number obtained by reversing the digits: zyx

Its numerical value = 100z + 10y + x

According to second information provided in the question,

(100x + 10y + z) + 198 = 100z + 10y + x

∴ 99z – 99x = 198

∴ z – x = 2

∴ z = x + 2 ....(2)

According to third information provided in the question,

x + z = y – 1

∴ x + x + 2 = y – 1 ....[from (2)]

∴ y = 2x + 3

Substituting the values of z and y in equation (1),

83x – 7(2x + 3) – 16(x + 2) = 0

∴ 83x – 14x – 21 – 16x – 32 = 0

∴ 53x – 53 = 0

∴ 53x = 53

∴ x = 1

∴ y = 2x + 3 = 2(1) + 3 = 2 + 3 = 5

∴ z = x + 2 = 1 + 2 = 3

Thus, the three-digit number is 153.