To determine the value of \( P \) in the number \( 465781P \) so that it is completely divisible by 12, we need to check the divisibility rules for both 3 and 4, since a number is divisible by 12 if it is divisible by both of these numbers.
### Step 1: Check divisibility by 3
The rule for divisibility by 3 states that the sum of the digits of the number must be divisible by 3.
**Calculation:**
- The digits of the number are \( 4, 6, 5, 7, 8, 1, P \).
- First, we calculate the sum of the known digits:
\[
4 + 6 + 5 + 7 + 8 + 1 = 31
\]
- Now, including \( P \), the total sum becomes:
\[
31 + P
\]
For \( 31 + P \) to be divisible by 3, we check the possible values of \( P \) (which can be from 0 to 9).
**Checking values:**
- If \( P = 0 \): \( 31 + 0 = 31 \) (not divisible by 3)
- If \( P = 1 \): \( 31 + 1 = 32 \) (not divisible by 3)
- If \( P = 2 \): \( 31 + 2 = 33 \) (divisible by 3)
- If \( P = 3 \): \( 31 + 3 = 34 \) (not divisible by 3)
- If \( P = 4 \): \( 31 + 4 = 35 \) (not divisible by 3)
- If \( P = 5 \): \( 31 + 5 = 36 \) (divisible by 3)
- If \( P = 6 \): \( 31 + 6 = 37 \) (not divisible by 3)
- If \( P = 7 \): \( 31 + 7 = 38 \) (not divisible by 3)
- If \( P = 8 \): \( 31 + 8 = 39 \) (divisible by 3)
- If \( P = 9 \): \( 31 + 9 = 40 \) (not divisible by 3)
From this, the possible values of \( P \) that make the sum divisible by 3 are \( 2, 5, \) and \( 8 \).
### Step 2: Check divisibility by 4
The rule for divisibility by 4 states that the last two digits of the number must form a number that is divisible by 4. The last two digits in our case are \( 1P \).
**Checking values:**
- If \( P = 0 \): Last two digits = 10 (not divisible by 4)
- If \( P = 1 \): Last two digits = 11 (not divisible by 4)
- If \( P = 2 \): Last two digits = 12 (divisible by 4)
- If \( P = 3 \): Last two digits = 13 (not divisible by 4)
- If \( P = 4 \): Last two digits = 14 (not divisible by 4)
- If \( P = 5 \): Last two digits = 15 (not divisible by 4)
- If \( P = 6 \): Last two digits = 16 (divisible by 4)
- If \( P = 7 \): Last two digits = 17 (not divisible by 4)
- If \( P = 8 \): Last two digits = 18 (not divisible by 4)
- If \( P = 9 \): Last two digits = 19 (not divisible by 4)
From this, the possible values of \( P \) that make the last two digits divisible by 4 are \( 2 \) and \( 6 \).
### Step 3: Find common values
The values of \( P \) that satisfy both conditions (divisible by 3 and 4) are:
- From divisibility by 3: \( 2, 5, 8 \)
- From divisibility by 4: \( 2, 6 \)
The only common value is \( P = 2 \).
### Conclusion
Thus, the value of \( P \) that makes \( 465781P \) completely divisible by 12 is:
\[
\boxed{2}
\]
