What are the divisibility short cuts that help you out?
Odd numbers can only be divided evenly by another odd number.
Even numbers can be evenly divided by either odd or even numbers.
2---Numbers that end in 0, 2, 4, 6, or 8 are evenly divisible by 2.
3---If the sum of a number's digits is evenly divisible by 3, the number is divisible by 3.
4---If the last two digits are both zero or they form a two digit number evenly divisible by 4, then
the whole number is evenly divisible by 4.
5--Any number ending in 5 or 0 is evenly divisible by 5.
6--If the number is evenly divisible by 2 and 3, it is divisible by 6.
If the number is even and the sum of the digits is evenly divisible by 3, the whole number is
divisible by 3.
7---Double the last digit and subtract from the number without the last digit. If the result is evenly
divisible by 7, so is the original number.
Multiply the left-hand digit by 3 and add the next digit. With the result, repeat as often as
necessary with the subsequent digits. If the final answer is divisible by 7, so is the original
number.
Divide the number into groups of 3 digits or thousands. Subtract the first group from the
second group. Subtract the third group from the result of the first subtraction. Continue
subtracting until the last group is subtracted. If the resulting number is divisible by 7, so is the
original number. This method works for 11 and 13 as well.
8---If the last 3 digits are zero or if they form a number that is evenly divisible by 8, then the whole
number is evenly divisible by 8.
9---If the sum of the digits is evenly divisible by 9, the number is evenly divisible by 9.
10--Any number that ends in 0 is evenly divisible by 10.
11--If all the digits in the number are the same and it has an even number of digits, the number is evenly
divisible by 11. An alternate method is to 1) add up the odd position digits, 2) add up the even position digits, and 3) calculate the difference between the two sums. If the difference is divisible by 11, the original number is divisible by 11.
Divide the number into groups of 3 digits or thousands. Subtract the first group from the
second group. Subtract the third group from the result of the first subtraction. Continue
subtracting until the last group is subtracted. If the resulting number is divisible by 11, so is the
original number.
12--If a number can be evenly divided by 3 and 4, then it can be evenly divided by 12.
13--Divide the number into groups of 3 digits or thousands. Subtract the first group from the
second group. Subtract the third group from the result of the first subtraction. Continue
subtracting until the last group is subtracted. If the resulting number is divisible by 13, so is the
original number.
15--If a number can be evenly divided by 3 and 5, then it can be evenly divided by 15.
17--Remove the last digit and multiply it by 5. Subtract the result from the rest of the number. Continue this process until you have a number smaller than 40. If this final result is divisible by 17, then the original number is divisible by 17.
18--If the number is evenly divisible by either 2 and 9 or 3 and 6, the number is divisible by 18.
Divisibility shortcuts can be helpful in quickly determining if a number is divisible by another number without performing the actual division process. Here are some common divisibility shortcuts:
1. Divisibility by 2: A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
2. Divisibility by 3: If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.
3. Divisibility by 4: If the last two digits of a number are either both zero or form a two-digit number that is divisible by 4, then the whole number is divisible by 4.
4. Divisibility by 5: Any number ending in 5 or 0 is divisible by 5.
5. Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
6. Divisibility by 7: There are two methods for testing divisibility by 7:
- Method 1: Double the last digit of the number and subtract it from the number formed by removing the last digit. If the result is divisible by 7, then the original number is also divisible by 7.
- Method 2: Multiply the leftmost digit by 3 and add the next digit to it. Repeat this process as needed with subsequent digits. If the final result is divisible by 7, then the original number is also divisible by 7.
7. Divisibility by 8: If the last three digits of a number are either all zero or form a number divisible by 8, then the whole number is divisible by 8.
8. Divisibility by 9: If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.
9. Divisibility by 10: Any number ending in 0 is divisible by 10.
10. Divisibility by 11: There are two methods for testing divisibility by 11:
- Method 1: If all the digits in the number are the same and the number has an even number of digits, then it is divisible by 11.
- Method 2: Add up the digits in odd positions and add up the digits in even positions. Find the difference between these two sums. If the difference is divisible by 11, then the original number is also divisible by 11.
11. Divisibility by 12: A number is divisible by 12 if it is divisible by both 3 and 4.
12. Divisibility by 13: Divide the number into groups of three digits or thousands. Subtract the first group from the second group, and subtract the third group from the result of the first subtraction. Continue subtracting until the last group is subtracted. If the resulting number is divisible by 13, then the original number is also divisible by 13.
13. Divisibility by 15: A number is divisible by 15 if it is divisible by both 3 and 5.
14. Divisibility by 17: Remove the last digit of the number and multiply it by 5. Subtract the result from the rest of the number. Continue this process until you have a number smaller than 40. If the final result is divisible by 17, then the original number is also divisible by 17.
15. Divisibility by 18: A number is divisible by 18 if it is divisible by either 2 and 9, or 3 and 6.
Remember that these divisibility shortcuts are not foolproof and may not be applicable for very large numbers. They are meant to provide a quick estimate of divisibility in most cases.