If the seven digit number 48649P2 is exactly divisible by 12, what is the value of the digit P?

Introduction / Context:
This question focuses on divisibility rules for the number 12 and how they restrict the possible value of a missing digit in a large number. It tests whether candidates know that a number is divisible by 12 if and only if it is divisible by both 3 and 4, and whether they can apply these tests to a number with an unknown digit.

Given Data / Assumptions:
We have the number 48649P2, where P is an unknown digit from 0 to 9.The number is divisible by 12.We must determine the correct value of P.

Concept / Approach:
A number is divisible by 12 if it is divisible by both 3 and 4. The divisibility rule for 4 states that the last two digits of the number must form a number divisible by 4. The rule for 3 says that the sum of the digits must be divisible by 3. We can use these two tests to deduce restrictions on P and then choose the value that satisfies both simultaneously.

Step-by-Step Solution:
Step 1: Apply the rule for divisibility by 4. The last two digits are P2, which form the number 10P + 2.Step 2: For divisibility by 4, 10P + 2 must be divisible by 4.Step 3: Note that 10P + 2 modulo 4 is the same as (2P + 2) modulo 4, since 10 is congruent to 2 modulo 4.Step 4: So 2P + 2 must be divisible by 4. That means 2P + 2 = 4k.Step 5: Rearranging, 2P = 4k − 2 = 2(2k − 1), so P = 2k − 1, which implies P is odd.Step 6: From the given options 1, 2, 3 and 4, the only odd candidates are 1 and 3.Step 7: Now apply the rule for divisibility by 3. The sum of the digits of 48649P2 must be divisible by 3.Step 8: Sum the known digits: 4 + 8 + 6 + 4 + 9 + 2 = 33. Adding P gives 33 + P.Step 9: For divisibility by 3, 33 + P must be a multiple of 3. Since 33 is already a multiple of 3, P itself must be a multiple of 3.Step 10: Among the candidate values 1 and 3, only 3 is a multiple of 3.Step 11: Therefore P = 3 satisfies both the divisibility by 4 and by 3, so the number 4864932 is divisible by 12.

Verification / Alternative check:
To verify, construct the number with P = 3, which gives 4864932. The last two digits 32 form a number divisible by 4. The sum of the digits is 4 + 8 + 6 + 4 + 9 + 3 + 2 = 36, which is divisible by 3. Hence 4864932 is divisible by 12. Choosing P = 1 instead would give 4864912; although 12 is divisible by 4, the digit sum 4 + 8 + 6 + 4 + 9 + 1 + 2 = 34 is not divisible by 3.

Why Other Options Are Wrong:
For P = 1, the number fails the divisibility by 3 test. For P = 2 or P = 4, the last two digits 22 and 42 respectively are not divisible by 4, so the number fails the divisibility by 4 test. Therefore those options cannot produce a number divisible by 12.

Common Pitfalls:
Some learners apply only the rule for 3 or only the rule for 4 and stop early, which may leave more than one possible value for P. Others miscalculate the digit sum or incorrectly handle the last two digits when testing for divisibility by 4. Using both rules carefully ensures a unique and correct answer.

Final Answer:
The value of the digit P that makes 48649P2 divisible by 12 is 3.