If the product 4864 × 9P2 is divisible by 12, what are the possible digit values of P?

Given data

• Number: 4864 × 9P2 must be divisible by 12.

Concept / Approach

• 12 = 3 × 4 with gcd(3, 4) = 1. The product must be divisible by both 3 and 4.
• 4864 is already divisible by 4 (last two digits 64). So the product is automatically divisible by 4.
• For divisibility by 3, at least one factor must be divisible by 3. Since 4864 is not (4+8+6+4=22), 9P2 must be divisible by 3.

Step-by-step check for divisibility by 3

Digit sum of 9P2 is 9 + P + 2 = 11 + P.We need 11 + P ≡ 0 (mod 3) ⇒ P ≡ 1 (mod 3).Thus P ∈ {1, 4, 7}.

Verification

9 1 2 → sum 12 ✔; 9 4 2 → sum 15 ✔; 9 7 2 → sum 18 ✔.

Common pitfalls

• Imposing divisibility by 4 on 9P2 unnecessarily; 4864 already ensures the product is divisible by 4.

Final Answer

Possible digits for P: 1, 4, 7.