Appendix D
TESTS FOR DIVISIBILITY
It is often useful to know whether a number is evenly divisible by another number. Here are some ways
you can find out without doing a full division.
Divisibility by 2: All even numbers are divisible by 2; that is, all whole numbers for which the final
digit is 0, 2, 4, 6 or 8.
Divisibility by 3: If the sum of the digits is divisible by 3, then the number is also divisible by 3. For
instance, the digit sum of 45 is 9 (4 + 5 = 9), which is divisible by 3, so 45 is divisible by 3.
Divisibility by 4: If the last two digits are divisible by 4, the number is divisible by 4. This is because
100 is evenly divisible by 4 (100 = 4 × 25).
There is an easy shortcut. Don’t worry how long the number is; if the tens digit is even, forget it and
check if the units digit is divisible by 4. (It must be zero, 4 or 8.) If the tens digit is odd, carry 1 to the
units digit and check if it is divisible by 4. That is because 20 is evenly divisible by 4 (4 × 5 = 20).
For instance, let’s check if 15,476 is divisible by four. The tens digit, 7, is odd, so we carry 1 to the
units digit, 6, to make 16. Sixteen is evenly divisible by 4, so 15,476 is evenly divisible by 4.
Is 593,768 divisible by 4? The tens digit, 6, is even so we can ignore it. The units digit, 8, is evenly
divisible by 4, so 593,768 is evenly divisible by 4.
Divisibility by 5: Five is easy. If the last digit of the number is 5 or 0, then the number is evenly
divisible by 5.
Divisibility by 6: If the number is even and the sum of the digits is divisible by 3, then the number is
also divisible by 6 (because 6 is 2 times 3). For instance, the digit sum of 54 is 9 (5 + 4 = 9), which is
divisible by 3, and 54 is an even number, so 54 is divisible by 6.
Divisibility by 7: You will have to read my book Speed Mathematics for a full explanation of this,
but here is a quick introduction. You multiply the final digit of the number by 5 and add the answer to
the number preceding it. If the answer is divisible by 7, then the number is divisible by 7. For example,
let’s take 343. The final digit is 3. We multiply 3 by 5 to get 15. We add 15 to the number in front of the
3, which is 34, to get 49. Then, 49 is evenly divisible by 7, so 343 is as well.
Divisibility by 8: If the final three digits are evenly divisible by 8 then the number is evenly divisible
by 8 (because 1,000 = 8 × 125).
Here we have an easy check similar to the check for 4. If the hundreds digit is even it can be ignored.
If it is odd, add 4 to the final two digits of the number. Then, if the final two digits are divisible by 8, the
number is divisible by 8.
For example, is the number 57,328 divisible by 8? The hundreds digit, 3, is odd so we add 4 to the
final two digits, 28. So, 28 + 4 = 32. Thirty-two is divisible by 8, so 57,328 is evenly divisible by 8.
Divisibility by 9: If the digit sum adds to 9 or is evenly divisible by 9, the number is divisible by 9.
Divisibility by 10: If the number ends with a 0, the number is divisible by 10.
Divisibility by 11: If the difference between the sum of the evenly placed digits and the oddly placed
digits is a multiple of 11, the number is evenly divisible by 11.
Divisibility by 12: If the digit sum is divisible by 3 and the last two digits are divisible by 4, the
number is divisible by 12.
Divisibility by 13: Again, a full explanation is in Speed Mathematics. The rule for 13 is similar to the