If the product 4864 x 9 P 2 is divisible by 12, then the value of P is:

Step-by-Step Solution

To determine the value of P such that the product (4864 * 9P2) is divisible by 12, we must apply divisibility rules. A number is divisible by 12 if it is divisible by both 3 and 4 (since 3 * 4 = 12 and they are coprime).

1. Check Divisibility by 4:

   • A number is divisible by 4 if its last two digits are divisible by 4.

   • In the product 4864 * 9P2, the first factor, 4864, is already divisible by 4 (as 64 is divisible by 4).

   • Therefore, the entire product will be divisible by 4 regardless of the value of P.

2. Check Divisibility by 3:

   • A number is divisible by 3 if the sum of its digits is divisible by 3.

   • The product 4864 * 9P2 must be divisible by 3. Since 4864 is not divisible by 3 (4 + 8 + 6 + 4 = 22, which is not a multiple of 3), the second factor, 9P2, must be divisible by 3 for the product to be divisible by 12.

3. Calculate P for 9P2 to be divisible by 3:

   • Sum of digits of 9P2 = 9 + P + 2 = 11 + P.

   • For 11 + P to be divisible by 3, P can be:

     • If P = 1, sum = 12 (divisible by 3).

     • If P = 4, sum = 15 (divisible by 3).

     • If P = 7, sum = 18 (divisible by 3).