I love math. Not just complex math, but arithmetic, too. I simply adore mathematical shortcuts. That’s why I spent the last couple of days finding shortcuts for divisibility tests:

• 3 - Sum of digits is divisible by 3.
• 4 - If the last two digits (tens and ones places) are divisible by 4. You can simplify this test by subtracting multiples of 2 from the tens place.
• 7 - Remove the digit in the ones place. Double it, and subtract that from the remaining number. Repeat until you have a one-digit number. If that number is 7, 0, or -7, the original number is divisible by 7.
• 8 - If the last three digits are divisible by 8. You can simplify this test by subtracting multiples of 2 from the hundreds place, and multiples of 4 from the tens place.
• 9 - Sum of digits is divisible by 9.
• 11 - If the alternating sum of the digits (first digit, minus second digit, plus third digit, etc) is divisible by 11.
• 13 - Remove the digit in the ones place. Multiply that digit by 4, and add that to the remaining number. Repeat until you have a number less than 40. If that number is 13, 26, or 39, the original number is divisible by 13.

The rules for 7 and 13 can easily be extended to just about any number under 100 (and, in fact, the rules listed here for 3, 9, and 11 are simple variants). As long as you understand how those ones work, you don’t even really need to memorize anything.