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.
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
To solve this problem, let's start by assigning the unknown number as a two-digit number in the form of "10a + b", where "a" represents the tens digit and "b" represents the units digit.
Given that the number divided by the sum of its digits has a quotient of 2 and a remainder of 2, we can write the equation:
(10a + b) / (a + b) = 2 + (2 / (a + b))
Similarly, we can write the equation for the product of the number and the sum of its digits:
(10a + b) * (a + b) = 112
Let's solve the first equation to find the values of "a" and "b".
(10a + b) / (a + b) = 2 + (2 / (a + b))
Multiplying both sides by (a + b) gives:
10a + b = 2(a + b) + 2
Expanding and simplifying further:
10a + b = 2a + 2b + 2
Rearranging the equation:
8a - b = 2
Now, let's solve the second equation.
(10a + b) * (a + b) = 112
Expanding and simplifying the equation:
10a^2 + 20ab + b^2 = 112
Rearranging the equation:
10a^2 + 20ab + b^2 - 112 = 0
To further solve the equations, we need to find the values of "a" and "b" that satisfy both equations. To do this, we'll use trial and error or a systematic approach by considering the range of possible values for "a" and "b".
Since the number is a two-digit number, "a" must be between 1 and 9 (inclusive), and "b" must be between 0 and 9 (inclusive). By trying different values of "a" and "b" within this range, we can find the values that satisfy both equations.
By trying different values, we find that when "a" equals 5 and "b" equals 2, both equations are satisfied:
8a - b = 40 - 2 = 38 ≠ 2
10a^2 + 20ab + b^2 - 112 = 500 + 200 + 4 - 112 = 592 ≠ 0
Therefore, there is no solution that satisfies both equations.