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?

(10a+b)= 2(a+b)+2

b = 8a-2

So, what choices are there?

a=1 b=6
a>1 : 8a-2 is too big for a single digit b.

So, 16/7 = 2 r 2

Thank You so much

Let's assume the two-digit integer is represented by "AB", where A represents the tens digit and B represents the ones digit.

According to the given information, the division of AB by the sum of its digits (A + B) gives a quotient of 2 and a remainder of 2.

So, we can write the equation as:

AB = 2(A + B) + 2

Now, let's simplify the equation:

AB = 2A + 2B + 2

Next, let's group like terms:

AB - 2A - 2B = 2

Factor out A from the first two terms and B from the next two terms:

A(B - 2) - 2(B - 1) = 2

Expand the equation:

AB - 2A - 2B + 2 = 2

Combine like terms:

AB - 2A - 2B = 0

Now, let's factorize:

A(B - 2) - 2(B - 2) = 0

Simplify again:

(A - 2)(B - 2) = 0

To find the two-digit integer, we need to find the values of A and B.

From the equation (A - 2)(B - 2) = 0, we know that either (A - 2) or (B - 2) must equal zero.

If (A - 2) = 0, then A = 2.
If (B - 2) = 0, then B = 2.

Since we are looking for a two-digit integer, both A and B cannot equal 2.

Therefore, there are no two-digit integers that satisfy the given conditions.

To find the two-digit integer, we'll need to translate the given information into an equation and solve for the unknown.

Let's suppose the two-digit integer is written as "10a + b," where 'a' represents the tens digit and 'b' represents the ones digit. Since it's a positive two-digit integer, both 'a' and 'b' would be greater than zero.

According to the problem statement, when we divide this number by the sum of its digits, the quotient is 2 with a remainder of 2. Mathematically, we can express this as:

(10a + b) ÷ (a + b) = 2 + (2 ÷ (a + b))

Simplifying further:

(10a + b) = 2(a + b) + 2

Now, let's solve this equation to find the values of 'a' and 'b.'

10a + b = 2a + 2b + 2

Subtracting 2a and 2b from both sides of the equation:

8a - b = 2

Now, let's list possible values for 'a' and 'b' that satisfy the equation:

a = 1, b = 6 (8 - 6 = 2)
a = 2, b = 4 (16 - 4 = 2)
a = 3, b = 2 (24 - 2 = 2)
a = 4, b = 0 (32 - 0 = 2) - (But since it's a two-digit integer, the value of 'b' must be non-zero)

Therefore, the two-digit integer satisfying the given conditions is 16.