How can you quickly know if one number will
divide evenly into another number, leaving no remainder? For example, will 3
divide evenly into 2,169,252? Well, I wouldn't have brought the subject up if I
didn't know some curious shortcuts...

**Division by 2**

No surprise here. Any number that ends in 0,2,4,6 or 8 is evenly divisible
by 2.

**Division by 3**

Add the number's digits. If the sum is evenly divisible by 3, then so is the
number. So, will 3 divide evenly into 2,169,252? Yes it will, because the sum of
the digits is 27, and 27 is divisble by 3. If you want, you can keep adding
numbers until one digit remains. For example, keep going with 27. 2 + 7 = 9,
which is also evenly divisible by 3.

**Division by 4**

If the number's last 2 digits are 00 or if they form a 2-digit number evenly
divisible by 4, then number itself is divisible by 4. How about 56,789,000,000?
Last 2 digits are 00, so it's divisible by 4. Try 786,565,544. Last 2 digits,
44, are divisible by 4 so, yes, the whole number is divisible by 4.

**Division by 5**

Any number that ends in a 0 or 5 is evenly divisible by 5. Easy enough.

**Division by 6**

The number has to be even. If it's not, forget it. Otherwise, add up the
digits and see if the sum is evenly divisible by 3. It it is, the number is
evenly divisible by 6. Try 108,273,288. The digits sum to 39 which divides
evenly into 13 by 3, so the number is evenly divisible by 6. If you want, you
can keep adding numbers until only one digit remains and do the same thing. So
in this case, 3 + 9 = 12 and 1 + 2 = 3, and 3 is evenly divisible by 3!

**Division by 7**

Multiply the last digit by 2. Subtract this answer from the remaining
digits. Is this number evenly divisible by 7? If it is, then your original
number is evenly divisible by 7. Try 364. 4, the last digit, multiplied by 2 =
8. 36, the remaining digits, minus 8 = 28. The last time I checked, 28 is evenly
divisble by 7, and thus, so is 364!

**Division by 8**

If the number's last 3 digits are 000 or if they form a 3-digit number
evenly divisible by 8, then the number itself is divisible by 8. How about
56,789,000,000? Last 3 digits are 000, so it's divisible by 8. Try 786,565,120.
The last 3 digits, 120, divide by 8 into 15, so yes, the whole number is
divisible by 8.

**Division by 9**

Sum the number's digits. If it divides by 9, you're in luck. As with the
tests for 3 and 6, you can keep adding numbers until you're left with only one
digit.

**Division by 10**

Any number that ends in 0 is evenly divisible by 10.

**Division by 11**

Here are four ways for different types of numbers:

- If
the sum of every other digit, starting with the first, is equal to the sum
of every other digit starting with the second, then the number is evenly
divisible by 11. Try 13057. 1+0+7 = 3+5, therefore it should divide evenly
by 11. And indeed it does: 13057 / 11 = 1187.
- If
all the digits are the same and there's an even number of digits, then the
number is evenly divisible by 11. 333,333 - Yes. 3,333,333 - No.
- If
the number is a 3 digit number with different digits, add the two outside
digits. If the difference between the sum and the middle digit is 11 then 11
divides evenly into the 3 digit number. If the sum is the same as the middle
digit, then 11 will also divide evenly into the number. Try 484. 4 + 4 = 8
which equals the middle digit so 11 divides into 484 evenly. How about 913?
9 + 3 = 12 and 12 - 1 = 11 so 913 is evenly divisible by 11.
- If
the digits are different, count 'em from the right and then add the numbers
in the odd positions and the even positions. Subtract the smaller number
from the larger. If the difference is evenly divisible by 11, so is your
original number. Take the number 181,907. The numbers 8,9, and 7 are in the
odd positions. They sum to 24. The numbers 1,1, and 0 are in the even
positions. They sum to 2. Subtract 2 from 24 to get 22. 22 divides by 11
into 2, so 181,907 is evenly divisible by 11.

**Division by 12**

If the number can be evenly divided by 3 and 4, the same can also be said
for 12. Use the methods for Division by 3 and Division by 4 above. If they both
work, your number is also evenly divisible by 12.

**Division by 15**

If the number can be evenly divided by 3 and 5, the same can also be said
for 15. Use the methods for Division by 3 and Division by 5 above. If they both
work, your number is also evenly divisible by 15.

**Division by 24**

If the number can be evenly divided by 3 and 8, the same can also be said
for 24. Use the methods for Division by 3 and Division by 8 above. If they both
work, your number is also evenly divisible by 24.

**Division by 33**

If the number can be evenly divided by 3 and 11, the same can also be said
for 33. Use the methods for Division by 3 and Division by 11 above. If they both
work, your number is also evenly divisible by 33.

**Division by 36**

If the number can be evenly divided by 4 and 9, the same can also be said
for 36. Use the methods for Division by 4 and Division by 9 above. If they both
work, your number is also evenly divisible by 36.

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