Divisibility Rules

You can tell that 9 goes into 7,452 without dividing. Add its digits: $7+4+5+2$ is 18. 9 goes into 18, so 9 goes into 7,452 too.

Why? 10 is $9+1$, and 100 is $99+1$. So 7,452 is some nines, plus the digits $7+4+5+2$. 9 goes into the nines. So only the digit sum (the digits added up) decides.

Quick tests, with no long division:

• 3 or 9 goes into a number when it goes into the digit sum.
• 2 goes into a number when the last digit is even.
• 5 goes in when it ends in 0 or 5. 10 goes in when it ends in 0.