Determining Values of Digits A and B in 3A54B10 for Divisibility by 330

To determine the values of the digits A and B in the number 3A54B10 such that it is divisible by 330, we first note that 330 2 × 3 × 5 × 11. Therefore, the number must be divisible by each of these factors. Let's break down the process.

Step-by-Step Divisibility Analysis

Step 1: Check Divisibility by 2

A number is divisible by 2 if its last digit is even. In this case, the last digit is 0, so 3A54B10 is divisible by 2.

Step 2: Check Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5. Since the last digit is 0, 3A54B10 is divisible by 5.

Step 3: Check Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits in 3A54B10 is: 3 A 5 4 B 1 0 A B 13. We need A B 13 to be divisible by 3.

Step 4: Check Divisibility by 11

A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11. The positions are:

Odd positions: 3, 5, B, 0 (1st, 3rd, 5th, 7th) Even positions: A, 4, 1 (2nd, 4th, 6th)

The sums are:

Sum of odd positions: 3 5 B 0 8 B Sum of even positions: A 4 1 A 5

The difference is: 8 B - (A 5) B - A 3. We need B - A 3 to be divisible by 11.

Summary of Conditions

A B 13 is divisible by 3. B - A 3 is divisible by 11.

Testing Values for A and B

Since A and B are digits (0-9), we can test possible values.

Case 1: Testing A 0

Let's test for A 0:

A B 13 B 13 to be divisible by 3 implies: B 13 equiv 0 mod 3 → B equiv 1 mod 3 → Possible values: 1, 4, 7.

B - A 3 B 3 to be divisible by 11 implies: B 3 equiv 0 mod 11 → B equiv 8 mod 11 → Possible value: 8.

Testing B 8:

3A54B10 3054810, which is divisible by 3: 3 0 5 4 8 1 0 21, and 21 is divisible by 3. 8 - 0 3 11, which is divisible by 11.

So, A 0 and B 8 works.

Case 2: Testing A 1

Let's test for A 1:

A B 13 1 B 13 B 14 to be divisible by 3 implies: B 14 equiv 0 mod 3 → B equiv 1 mod 3 → Possible values: 1, 4, 7.

B - A 3 B - 1 3 B 2 to be divisible by 11 implies: B 2 equiv 0 mod 11 → B equiv 9 mod 11 → Possible value: 9.

Testing B 9:

3A54B10 3154910, which is not divisible by 3: 3 1 5 4 9 1 0 23, and 23 is not divisible by 3.

Case 3: Testing A 2

Let's test for A 2:

A B 13 2 B 13 B 15 to be divisible by 3 implies: B 15 equiv 0 mod 3 → B equiv 0 mod 3 → Possible values: 0, 3, 6, 9.

B - A 3 B - 2 3 B 1 to be divisible by 11 implies: B 1 equiv 0 mod 11 → B equiv 10 mod 11 → Not possible since B must be a digit.

Continuing this process for other values, we conclude there are no other valid solutions.

Conclusion

The values of the digits A and B are:

A 0 B 8

Thus, 3A54B10 3054810 is divisible by 330.