The 7-digit numbers 74A52B1 and 326AB4C are each multiples of 3. Which of the following could be the value of C ?
None of the above.
To determine the possible values of C, we need to understand the divisibility rule for 3. A number is divisible by 3 if the sum of its digits is divisible by 3.
For the number 74A52B1, we already know it is divisible by 3. Therefore, the sum of its digits must be divisible by 3. The sum of the digits in this case is given by 7 + 4 + A + 5 + 2 + B + 1 = 19 + A + B. For this sum to be divisible by 3, the sum 19 + A + B must be divisible by 3. We can express this as 19 + A + B ≡ 0 (mod 3).
For the number 326AB4C, we also know it is divisible by 3. Therefore, the sum of its digits must be divisible by 3. The sum of the digits in this case is given by 3 + 2 + 6 + A + B + 4 + C = 15 + A + B + C. For this sum to be divisible by 3, the sum 15 + A + B + C must be divisible by 3. We can express this as 15 + A + B + C ≡ 0 (mod 3).
To find the possible values of C, we need to find values of C that satisfy both equations.
Let's analyze the possible values of C:
- If C = 0, then the equation 15 + A + B + C ≡ 0 (mod 3) becomes 15 + A + B ≡ 0 (mod 3), which means A + B ≡ 0 (mod 3).
- If C = 1, then the equation 15 + A + B + C ≡ 0 (mod 3) becomes 16 + A + B ≡ 0 (mod 3), which means A + B ≡ 2 (mod 3).
- If C = 2, then the equation 15 + A + B + C ≡ 0 (mod 3) becomes 17 + A + B ≡ 0 (mod 3), which means A + B ≡ 1 (mod 3).
From the above analysis, we can see that C can be 0, 1, or 2. These are the possible values for C.