The difference between my digits is 7. When you divide me by the sum of my digits, the quotient is the sum of my digits. That makes me a perfect you know what! Which number am I?
To find the number, let's break down the given information.
1. The difference between the digits is 7: Let's assume the two digits of the number are x and y, where x is the larger digit. So, x - y = 7.
2. When you divide the number by the sum of its digits, the quotient is the sum of its digits: This means that the number, let's call it xy, should satisfy the equation xy / (x + y) = x + y.
Now, let's solve these equations step by step:
From the first equation, x - y = 7, we can rewrite it as x = 7 + y.
Substituting this value of x in the second equation, we get:
(7 + y)y / (7 + y + y) = 7 + y
Simplifying further:
(7 + y)y / (7 + 2y) = 7 + y
Cross-multiplying:
(7 + y)y = (7 + y)(7 + 2y)
Expanding:
7y + y^2 = 49 + 7y + 14y + 2y^2
Rearranging and simplifying:
y^2 - 21y + 49 = 0
Now, we can solve this quadratic equation.
Factoring is convenient for this equation:
(y - 7)(y - 7) = 0
So, y = 7.
Substituting this value of y into x = 7 + y, we get x = 7 + 7 = 14.
Therefore, the number that satisfies the given conditions is 14.