Question

If a number of 2 digits is divided by the sum of its digits, the quotient is 2 and the remainder is 2. If it's multiplied by the sum of its digits, the result is 112. Find the number.

Answer 1

let the tens digit be x, let the unit digit be y
the number is 10x+y

(10x+y)/(x+y) = 2 + 2/(x+y)
times (x+y)
10x+y = 2x+2y + 2
8x - y = 2 ----> y = 8x-2

(10x+y)(x+y) = 112
10x^2 + 11xy + y^2 = 112
sub in y = 8x-2

10x^2 + 11(8x-2) + (8x-2)^2 = 112
10x^2 + 88x - 22 + 64x^2 - 32x + 4 - 112 = 0
74x^2 + 56x - 130 = 0
37x^2 + 28x - 65 = 0
x = (-28 ± √10404)/74
= 1 or some negative

if x = 1 , then y = 6

the number is 16

check: the sum of the digits is 7
16/7 = 2, remainder 2
16(7) = 112

My answer is correct