If a positive two-digit integer is divided by the sum of its digits, the quotient is 2 with a remainder of 2. What is the two-digit integer?
Isabelle/Clary -- why did you post this after you'd received an answer from one of our volunteer tutors?
http://www.jiskha.com/display.cgi?id=1483896126
(10T+1N)/(T+N) = 2 + 2/(T+N)
or
10T+N=2T+2N+2
8T-N=2
T=1, N=6 will work in that last equation
and check does 16 work in the problem statement (It does)
wish I had seen it, it wasted my time doing it again.
To find the two-digit integer, let's say it is represented by the variables "AB," where A is the tens digit and B is the units digit. We know that dividing AB by the sum of its digits gives a quotient of 2 with a remainder of 2.
So, we can set up the equation:
AB = 2(A + B) + 2
Expanding the right side, we get:
AB = 2A + 2B + 2
Next, we simplify the equation:
AB - 2A - 2B = 2
Now, let's factor out A and B:
A(B - 2) - 2(B - 2) = 2
(A - 2)(B - 2) = 2
Since we are looking for a two-digit number, A ≠ 0. Therefore, we have two possible pairs:
Case 1: A - 2 = 1 and B - 2 = 2
In this case, we have A = 3 and B = 4. Therefore, the two-digit integer is 34.
Case 2: A - 2 = 2 and B - 2 = 1
In this case, we have A = 4 and B = 3. Therefore, the two-digit integer is 43.
Thus, the possible two-digit integers that satisfy the given conditions are 34 and 43.