A number is divisible by another number if after dividing; the remainder is '0'. For example, 18 is divisible by 3 because 18 ÷ 3 = 6 with 0 remainder. However, 25 is not divisible by 4 because 25 ÷ 4 = 6 with a remainder of 1.
There are several mental math tricks that can be used to find the remainder after division without actually having to do the division. Here are some curious shortcuts to quickly know if one number will divide evenly into another number, leaving no remainder.
Divisibility by 2
No surprise here. A number is divisible by 2 if the last digit is ‘0’ or even. That is any number that ends in 0, 2, 4, 6 or 8 is evenly divisible by 2.
Divisibility by 3
A number is divisible by 3 if the sum of all the digits is divisible by 3.
Example: 34,164 is divisible by 3 because the digit sum of the number is (3+4+1+6+4=) 18, which is divisible by 3. If you want, you can keep adding numbers until one digit remains. For example, keep going with 18. 1+8=9; this is also evenly divisible by 3.
Another example: 87687687. The digits add up to 57, and 5 plus 7 is 12, so the original number is divisible by 3.
Divisibility by 4
If the number’s last 2-digits are ‘00’ or if they form a 2-digit number evenly divisible by 4, then the number itself is divisible by 4.
Example: 34,164 is divisible by 4 because last 2-digits i.e. 64 is divisible by 4.
Again, 1732784864826421834612 is divisible by 4 also, because last 2-digits of the number which is 12 is divisible by 4.
Divisibility by 5
Any number that ends in a 0 or 5 is evenly divisible by 5. Easy enough!
Divisibility by 6
A number is divisible by 6 if it is even and divisible by 3.
Example: 34,164 is divisible by 6 because it is even and divisible by 3. Another Example: 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!
Divisibility by 7
To find out if a number is divisible by 7, take the last digit, double it, and subtract it from the rest number. If this number is evenly divisible by 7; then the original number is evenly divisible by 7. This process may be repeated if the result is too large for simple inspection of divisibility of 7.
If you had 203, you would double the last digit to get 6, and subtract that from 20 to get 14. If you get an answer divisible by 7 (including zero); then the original number is divisible by 7.
If you don't know the new number's divisibility, you can continue this pattern until you find a number you know is or is not divisible by 7.
Example: 7203 is divisible by 7 because
a) 2 × 3 = 6.
b) 720 – 6 = 714 which is divisible by 7.
14443 is not divisible by 7 because
a) 3 × 2 = 6.
b) 1444 – 6 = 1438.
c) 8 × 2 = 16.
d) 143 – 16 = 127 which is not divisible by 7.
Another method:
1. Write down just the digits in the tens and ones places.
2. Take the other numbers to the left of those last 2-digits, and multiply them by 2.
3. Add the answer from step two to the number from step one.
4. If the sum from step three is divisible by 7, then the original number is divisible by 7 as well. If the sum is not divisible by 7, then the original number is not divisible by 7.
For example, if the number we are testing is 413, then
5. Write down just the digits in the tens and ones places: 13.
6. Take the other numbers to the left of those last two digits, and multiply them by 2: 4 × 2 = 8.
7. Add the answer from step two to the number from step one: 13 + 8 = 21.
8. 21 is divisible by 7. Therefore, our original number, 413 is also divisible by 7.
Note: This method takes a lot of practice and is sometimes easier to just work it out individually.
A number is divisible by 8 if the number's last 3 digits are ‘000’ or if they form a 3-digit number evenly divisible by 8.
Example: 34,168 is divisible by 8 because last 3 digits i.e. 168 is divisible by 8.
How about 56,789,000,000? Last 3 digits are 000, so it's divisible by 8.
Again 33333256 is divisible by 8; 33333258 is not.
How can you tell whether the last three digits are divisible by 8?
If the first digit is even, the number is divisible by 8 if the last two digits are. If the first digit is odd, subtract 4 from the last two digits; the number will be divisible by 8 if the resulting last two digits are. So, to continue the last example, 33333256 is divisible by 8 because the digit in the hundreds place is an even number, and the last two digits are 56, which is divisible by 8. Again, 33333258 is not divisible by 8 because (though) the digit in the hundreds place is an even number, but the last two digits are 58, which is not divisible by 8.
Divisibility by 9
A number is divisible by 9 if the sum of the digits is divisible by 9.
As with the tests for 3 and 6, you can keep adding numbers until you're left with only one digit.
Example: 34,164 is divisible by 9 because the digit sum is (3+4+1+6+4 =) 18, which is divisible by 9.
Divisibility by 10
Any number that ends in 0 is evenly divisible by 10.
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