In the three-digit number 4P6 the letter P stands for a single digit. The whole number divides exactly by 9. What is P?

A number divides exactly by 9 when its digits add to a multiple of 9. Here the digits add to $4+P+6=10+P$, and as $P$ runs from 0 to 9 that total runs from 10 to 19, in which the only multiple of 9 is 18. So $10+P=18$ and $P=8$, giving $486=9\times 54$. The traps: 2 makes the digits add to 12 and 5 makes them add to 15 — both divide by 3, which is the rule people half-remember, but neither divides by 9; and 9 is the guess that the missing digit must simply copy the divisor.