Find digits A and B in the number below so the folling condition are true. The 5-digit number must be divisible by 4. The 5-digit number must be divisible by 9. Digit A cannot be the same as Digit B.



12A3B

Explain the steps you followed to solve the problem.

If a number is divisible by 9, the sum of its digits is too. That is, they must add up to a multiple of 9.

6+A+B = 9k
so,
A+B = 9k-6
The most they can be is 18, so they must add up to 12 or 3.

3B must be a multiple of 4, since xxx00 is. So, 3B must be 32 or 36. Can't be 36, because that would make B=6 so A=6.

So, B = 2 and A=1

12132 = 4*3283 = 9*1348

To find the digits A and B in the number 12A3B such that the number is divisible by 4 and 9, and digit A cannot be the same as digit B, follow these steps:

Step 1: Divisibility by 4
A number is divisible by 4 if the last two digits of the number are divisible by 4. In this case, the number is 12A3B, so the last two digits are 3B. To make the number divisible by 4, B must be an even digit since all even digits are divisible by 4. The possible even digits are 0, 2, 4, 6, and 8.

Step 2: Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The number 12A3B must be divisible by 9, so we need to find A such that the sum of the digits is divisible by 9. The sum of the digits 1 + 2 + A + 3 + B should be divisible by 9. We already know that B is an even digit, so it cannot contribute to the sum for divisibility by 9. Therefore, we only need to focus on the digits 1, 2, A, and 3. To make the sum divisible by 9, A should be chosen such that (1 + 2 + A + 3) is divisible by 9. The possible values for A are 3, 6, and 9, as these values will make the sum divisible by 9.

Step 3: Digit A cannot be the same as Digit B
The condition states that digit A cannot be the same as digit B. We have determined that B should be an even digit, and the possible even digits for B are 0, 2, 4, 6, and 8. We also found that A can take the values 3, 6, and 9. Checking all the possible combinations, we can see that the digit A cannot be equal to any of the even digits since those are possibilities for B. Therefore, we can conclude that A cannot be any of the even digits 0, 2, 4, 6, or 8.

By following these steps, we have determined that A cannot be an even digit, so A must be either 3, 6, or 9. Meanwhile, B can be any of the even digits since A cannot be the same as B.