Calculations - Divisibility Rules - Learning

If you don't follow rules in Math, you'll get it wrong!

Calculations - Divisibility Rules - Learning

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Ah, we’re back to division. That should seem rather an easy math task by now and, in reality, it is. However, up until now the focus has been on your ability to understand and learn how to do the process each step of the way. That part won’t end - BUT - did you know that there are some very basic divisibility rules that can help you solve a problem even faster? Well, it’s time to let you in on these little secrets!

Although there are a number of divisibility rules, there are only 10 basic and easy rules to remember. These 10 basic rules of divisibility are as follows:

Rule 1: All numbers are evenly divisible by the number “1”. For example, the number 78 when divided by “1” equals “78”.

The number 43 when divided by the number “1” will equal “43”. The number 999 when divided by the number “1” will equal “999” and so on.

Rule 2: If a number ends with a 0, 2, 4, 6 or 8 then that number, no matter how long it is, will be evenly divisible by the number “2”. For example, look at the number 318. The last number or digit is an “8” so this tells us that the number can be evenly divided by the number “2” so let’s see if it works. 318 ÷ 2 = 159 - It worked!

Rule 3: If the combined sum of the number is divisible by the number 3, then the entire number is evenly divisible by the number “3”. For example, let’s look at the same number above, i.e., 318. Now let’s find the sum of the number so 3 + 1 + 8 = 12. The number 12 is divisible by 3 in that 12 ÷ 3 = 4. This then tells us that the number 318 is evenly divisible by the number “3” so let’s see if it works. 318 ÷ 3 = 106 - It worked!

Rule 4: If the last two digits of a number are divisible by the number “4” then the entire number is evenly divisible by the number “4”. For example, let’s look at the number 5236. The last two digits are “36” and are divisible by 4 as follows: 36 ÷ 4 = 9. Now we should be able to evenly divide the entire number by “4” so: 5236 ÷ 4 = 1309 - It worked!

Rule 5: If the last digit of a number is either a 0 or a 5, then the entire number is evenly divisible by the number “5”. For example, let’s look at the number 935. The last digit is a “5” so the number should be evenly divisible by the number “5” so 935 ÷ 5 = 187 - It worked!

Rule 6: If the last digit of a number is divisible by the number “2” and the sum of all the numbers is divisible by the number “3” then the entire number can be evenly divisible by the number “6”. Okay, let’s see how this works. Take the number 6294. The last digit is the number “4” and it is divisible by the number “2”, i.e., 4 ÷ 2 = 2, and the sum of the numbers 6 + 2 + 9 + 4 = 21 and 21 is divisible by the number “3”, i.e., 21 ÷ 3 = 7, then the entire number 6294 should be evenly divisible by the number “6” so let’s see if it is. 6294 ÷ 6 = 1049 - It worked!

Rule 7: If the sum of all of the digits is evenly divisible by the number “9” then the entire number is evenly divisible by the number “9”. For example, let’s look at the number 6687. Now we need to find the sum of the numbers so 6 + 6 + 8 + 7 = 27. The number “27” is divisible by the number “9” as 27 ÷ 9 = 3. This tells us that 6687 should be evenly divisible by the number “9” so let’s see if it works. 6687 ÷ 9 = 743 - It worked!

Rule 8: If the last digit of a number is “0” then the entire number is evenly divisible by the number “10”. For example, let’s take the number 126490. The last digit is a “0” so it should be evenly divisible by the number “10” so let’s see if it is. 126490 ÷ 10 = 12649 - It worked!

Rule 9: This one is a little tricky but if you have a number such as 4191 and, moving left to right, you subtract every second number from its preceding number and then you add the remaining numbers, if their sum is divisible by the number “11”, then the entire number is evenly divisible by the number “11”. Let’s break this down so you can see how it works. Again, the number is 4191 so we subtract every second number from the number that precedes it so: 4 - 1 = 3 and 9 - 1 = 8 then 3 + 8 = 11 and “11” is divisible by “11” as 11 ÷ 11 = 1. Now the entire number 4191 should be evenly divisible by the number “11” as 4191 ÷ 11 = 381 - It worked!

Rule 10: If the number is evenly divisible by both the numbers “3” and “4” (see Rules 3 and 4 to work this out), then the entire number will be evenly divisible by the number “12”. For example, let’s look at the number 5136. Rule 3 tells us that if the combined sum of the numbers is divisible by the number “3” then the entire number is evenly divisible by the number “3” so that gives us 5 + 1 + 3 + 6 = 15. The number 15 is divisible by the number “3” as 15 ÷ 3 = 5. Now Rule 4 tells us that if the last two digits of the number are a multiple of the number “4” then the entire number is evenly divisible by the number “4”. The last two digits are 36 and we know that 36 ÷ 4 = 9. Since our number 5136 can evenly be divisible by the numbers “3” and “4” then is should be evenly divisible by the number “12” so let’s check that out. 5136 ÷ 12 = 428 - It worked!

Take a few moments to read over these rules a few times. When you think you’ve got them done, move on to the following ten questions and see if you can find the right answer without looking back at the rules.

1.
Which of the following numbers are evenly divisible by the number “1”?
1397
2854
5277
Each one is evenly divisible by the number “1”
Per Rule 1, all numbers are evenly divisible by the number “1”. Answer (d) is correct
2.
Which of the following numbers is evenly divisible by the number “2”?
497
596
683
Each one is evenly divisible by the number “2”
Answer (c) ends in one of these numbers so they are not evenly divisible by the number “2”. Answer (b) does end with the number “6” and 596 ÷ 2 = 298. Answer (b) is the correct answer
3.
Which of the following numbers is evenly divisible by the number “3”?
397
853
624
Each one is evenly divisible by the number “3”
Rule 3 tells us that if the combined sum of the number is divisible by the number 3, then the entire number is evenly divisible by the number “3”. Answer (a) would then be 3 + 9 + 7 = 19. The number 19 is not evenly divisible by the number “3” so it is not the correct answer. Answer (b) would be 8 + 5 + 3 = 16 and 16 is not evenly divisible by the number “3”. Answer (c) would be 6 + 2 + 4 = 12. The number 12 is evenly divisible by the number “3” as 12 ÷ 3 = 4. Answer (c) is the correct answer
4.
Which of the following numbers is evenly divisible by the number “4”?
1276
1078
934
Each one is evenly divisible by the number “4”
Rule 4 tells us that if the last two digits of a number are divisible by the number “4” then the entire number is evenly divisible by the number “4”. The last two digits of Answer (b) are 78 and these are not evenly divisible by the number “4” as 78 ÷ 4 = 19.5. The last two digits of Answer (c) are 34 and this number is also not evenly divisibly by the number “4” as 34 ÷ 4 = 8.5. The last two digits of Answer (a) are 76 and this is evenly divisible by the number “4” as 76 ÷ 4 = 19. Now let’s see if the entire number is evenly divisible by the number “4” so 1276 ÷ 4 = 319. Answer (a) is the correct answer
5.
Which of the following numbers is evenly divisible by the number “5”?
5576
1862
3945
Each one is evenly divisible by the number “5”
Rule 5 tells us that if the last digit of a number is either a 0 or a 5, then the entire number is evenly divisible by the number “5”. The last digit on Answer (a) and Answer (b) is not a 0 or a 5 so they are not evenly divisible by the number “5”. Answer (c) does end in the number “5” so let’s see if the entire number is evenly divisible by the number “5”. 3945 ÷ 5 = 789 - Answer (c) is the correct answer
6.
Which of the following numbers is evenly divisible by the number “6”?
2954
4338
5162
Each one is evenly divisible by the number “6”
Rule 6 tells us that if the last digit of a number is divisible by the number “2” and the sum of all the numbers is divisible by the number “3” then the entire number can be evenly divisible by the number “6”. The last digit of Answers (a), (b) and (c) are each divisible by the number “2” so now we have to see if the sum of each number is divisible by the number “3”. Answer (a) 2 + 9 + 5 + 4 = 20. The number 20 is not evenly divisible by the number “3”. Answer (c) 5 + 1 + 6 + 2 = 14. The number 14 is not evenly divisible by the number “3”. Answer (b) 4 + 3 + 3 + 8 = 18. The number 18 is divisible by the number “3” in that 18 ÷ 3 = 6. Answer (b) is the correct answer
7.
Which of the following numbers is evenly divisible by the number “9”?
4536
8289
9495
Each one is evenly divisible by the number “9”
Rule 7 tells us that if the sum of all of the digits is evenly divisible by the number “9” then the entire number is evenly divisible by the number “9”. Answer (a) is 4 + 5 + 3 + 6 = 18. The number 18 is evenly divisible by the number “9” in that 18 ÷ 9 = 2. Answer (b) is 8 + 2 + 8 + 9 = 27. The number 27 is evenly divisible by the number “9” in that 27 ÷ 9 = 3. Answer (c) is 9 + 4 + 9 + 5 = 27 and we already know that 27 is evenly divisible by the number “9” so Answer (d) is the correct answer
8.
Which of the following numbers is evenly divisible by the number “10”?
12896
14730
16398
Each one is evenly divisible by the number “10”
Rule 8 tells us that if the last digit of a number is “0” then the entire number is evenly divisible by the number “10”. With Answers (a) and (c), neither of these numbers end with a “0” so they are not evenly divisible by the number “10”. Answer (b) does end in a “0” so let’s see if it is evenly divisible by the number “10” so that 14730 ÷ 10 = 1473. Answer (b) is the correct answer
9.
Which of the following numbers is evenly divisible by the number “11”?
9471
6384
7283
Each one is evenly divisible by the number “11”
Rule 9 tells us that if you have a number and, moving left to right, you subtract every second number from its preceding number and then you add the remaining numbers, if their sum is divisible by the number “11”, then the entire number is evenly divisible by the number “11”. Starting with Answer (b) you have 6 - 3 = 3 and then 8 - 4 = 4 which then gives you 3 + 4 = 7 and the number “7” is not evenly divisible by the number “11” so it is not the correct answer. Answer (c) gives us 7 - 2 = 5 and 8 - 3 = 5 which then gives you 5 + 5 = 10 and the number “10” is not evenly divisible by the number “11” so it is not the correct answer. Finally, Answer (a) gives us 9 - 4 = 5 and 7 - 1= 6 which then gives us 5 + 6 = 11 and the number “11” is evenly divisible by the number “11” in that 11 ÷ 11 =1 making Answer (a) the correct answer
10.
Which of the following numbers is evenly divisible by the number “12”?
2896
4179
8124
Each one is evenly divisible by the number “12”
Rule 10 tells us that if the number is evenly divisible by both the numbers “3” and “4” then the entire number will be evenly divisible by the number “12”. In Answer (a) the last two digits are evenly divisible by the number “4” in that 96 ÷ 4 = 24, however the sum of the number being 2 + 8 + 9 + 6 = 25, the number “25” is not evenly divisible by the number “3” so it is not correct. In Answer (b) the sum of the number is 4 + 1 + 7 + 9 = 21 and the number “21” is evenly divisible by the number “3” in that 21 ÷ 3 = 7, however the last two digits of 79 are not evenly divisible by the number “4” so Answer (b) is also not correct. In Answer (c) the sum of the number is 8 + 1 + 2 + 4 = 15 and the number “15” is evenly divisible by the number “3” in that 15 ÷ 3 = 5 and then the last two digits of 24 is evenly divisible by the number “4” in that 24 ÷ 4 = 6. As this number is evenly divisible by both “3” and “4” the entire number should now be evenly divisible by the number “12” so 8124 ÷ 12 = 677. Answer (c) is the correct answer
Author:  Christine G. Broome

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