In the multiplication probelem at the right,A,B,C represent different non-zero digits. Each time the same letter appears it represents the same digit. Find the 2 numbers represnted by AA and BC.(0 is zero).

A A
x B C
------
B B O
A A
-----
6 6 0

If A*C is 10, then a or c is 2,5

which means BB is 11
Because A+B is 6 without a carry, then
if A is 5
AA=55
BB=11
BC=12
Your teacher is too easy.

To find the numbers represented by AA and BC in the multiplication problem, we will need to use a systematic approach by using logical deductions.

Let's analyze the problem step by step:

1. From the ones column: Since BC multiplied by A gives the ones column's digit 0, it means that C multiplied by A must be equal to 0. However, we also know that A cannot be zero. Therefore, we can deduce that C must be 0.

2. The next step is from the tens column: According to the problem, AA multiplied by B gives the tens column's digit B. We can determine which numbers for A and B satisfy this equation.

- A multiplied by B results in a product that has two digits (AA), so we know that B must be at least 2.

- B multiplied by B results in a product with one digit (B). The only numbers that satisfy this condition are 2 and 3 since 2 * 2 = 4 and 3 * 3 = 9.

Therefore, B can only be 2 or 3.

3. Let's consider B = 2 first:
When B = 2, we know that A * 2 = B. Since B = 2, A must be 1. So, AA = 11.
Substituting these values into the equation:

11
x 2 0
------
0 --> This is not possible since A * B gives 0 in the ones column.
1 1 --> This is not possible since the multiplication of one-digit numbers cannot result in a two-digit number.

4. Let's try B = 3 now:
When B = 3, we know that A * 3 = B. Since B = 3, A must be 1. So, AA = 33.
Substituting these values into the equation:

33
x 3 0
------
09
09
------
990

In this case, the equation is satisfied, and the result is 990. Therefore, AA represents the number 33, and BC represents the number 90.

So, the numbers represented by AA and BC are 33 and 90, respectively.