If the product of 4856 x 91P is divisible by 12, where 91P is a 3 digit…

To determine the least value of P such that the product 4856 × 91P is divisible by 12, we must analyze the divisibility rules for 12. A number is divisible by 12 if and only if it is divisible by both 3 and 4.

Step-by-Step Analysis
Analyze 4856:

Divisibility by 4: A number is divisible by 4 if its last two digits are divisible by 4. The last two digits of 4856 are 56. Since 56 / 4 = 14, 4856 is divisible by 4.

Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 4856 is 4 + 8 + 5 + 6 = 23. Since 23 is not divisible by 3, 4856 is not divisible by 3.

Determine Requirements for the Product (4856 × 91P):

Since 4856 is already divisible by 4, the product 4856 × 91P will be divisible by 4 regardless of the value of 91P.

Because 4856 is not divisible by 3, for the product to be divisible by 3, the second number, 91P, must be divisible by 3.

Solve for P:

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

The sum of the digits of 91P is 9 + 1 + P = 10 + P.

We need 10 + P to be divisible by 3.

Testing the smallest possible digits for P:

If P = 0, sum = 10 (not divisible by 3)

If P = 1, sum = 11 (not divisible by 3)

If P = 2, sum = 12 (divisible by 3)

The least value of P that makes 91P divisible by 3, and thus the entire product divisible by 12, is 2.