Understanding the Divisibility Rule for 4

To determine if a number is divisible by 4, we only need to look at the number formed by its last two digits. If the number formed by the last two digits is divisible by 4, then the entire number is divisible by 4.

Applying the Rule to 87893P4

In the given 7-digit number, $87893P4$, the last two digits form the number $P4$. Here, 'P' represents a single digit from 0 to 9.

According to the divisibility rule for 4, the number $87893P4$ will be divisible by 4 if and only if the number $P4$ is divisible by 4.

Finding the Largest Digit for P

We need to find the largest possible digit that can replace P such that the two-digit number $P4$ is divisible by 4. Let's test the possible digits for P, starting from the largest (9) downwards:

• If P = 9: The number formed is 94. Is 94 divisible by 4? $94 \div 4 = 23.5$. No, 94 is not divisible by 4.
• If P = 8: The number formed is 84. Is 84 divisible by 4? $84 \div 4 = 21$. Yes, 84 is divisible by 4.

Since we are looking for the largest number that should replace P, and we found that P=8 works, this is our answer. We don't need to check smaller digits like 2 or 0, although they would also make the number divisible by 4 (forming 24 and 04 respectively).

Conclusion

The largest digit that can replace P in the number $87893P4$ to make it divisible by 4 is 8.