CONNECT: Divisibility

Transcription

1 CONNECT: Divisibility If a number can be exactly divided by a second number, with no remainder, then we say that the first number is divisible by the second number. For example, 6 can be divided by 3 so we say that 6 is divisible by 3. (We can also say that 3 is a factor of 6 and 6 is a multiple of 3, but that is not what this handout is about!) It can be difficult to divide numbers which are large or perhaps unfamiliar to you. However, there are procedures that enable us to quickly work out whether a number is divisible by other numbers. (The methods don t give the answer for the division problem, they are simply a guide to see if the number CAN be divided by the other number.) This is useful for simplifying fractions, or in a division problem to check if your answer should have a remainder or not. You need to know that the word digit refers to symbols for numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, the number contains the digits 2, 3, 4, 5, 7 and 8. We can add the value of each of those digits together or do other maths operations on them. So, if we add all the digits in , we get , and that gives 29. If we add the digits in , we get , which is 21. Divisibility Tests 2 If a number is even (that is, if its last digit is 2, 4, 6, 8, or 0), it can definitely be divided by 2. For a number that is as large as it is now easy to see that we can divide it exactly by 2 (because it ends in 6). 3 To work out whether a number can be divided by 3, add up all the digits in the number. If that total is divisible by 3, then the original number is also divisible by 3. Our example above, , can be divided by 3, because when you add its digits, you get 36 ( =36) which can be exactly divided by 3. (To calculate what the value is when is divided by 3, we then need to do the division procedure, but at least we know there should not be any remainder!) 4 To work out whether a number can be divided by 4, if its last two digits, when read as a two-digit number, can be divided by 4, then the whole number can be divided by 4. For example, we know straight away that CAN be divided by 4 because 16 can! A fast way to divide by 4 is to divide the 1

2 original number by 2, and then by 2 again. (You should get when you divide by 4). For our example above, however, , we see that the last two digits together are 06 (which is the same as 6) and 6 CANNOT be exactly divided by 4, so don t bother trying to divide 4 into it (or, if you do, you should get a remainder!). 5 If the number you are trying to divide ends with 0 or 5, then it is divisible by 5. If it ends in anything else, you will have a remainder when you divide. So, we can immediately tell that is divisible by 5. Of course we would then have to do the division to obtain the answer, but at least we know there will be no remainder when we divide. 6 If a number can be divided by both 2 and 3, then it is divisible by 6. So, if a number is even and if the total of its digits can be divided by 3, then the original number is divisible by 6. This means that our number from above, , MUST be divisible by 6 because it is both even and the sum of its digits (36) is divisible by 3. Let s do the division: ) (Note that there is no remainder. So the rule, that the number is both even and divisible by 3, works.) 7 Unfortunately 7 is not so straightforward, however, give this a try! To work out if a number is divisible by 7, take the last digit off the number, double that digit and subtract the doubled number from the remaining number. If the result is evenly divisible by 7 it may be negative then the number is divisible by seven. This may need to be repeated several times. Example: Is 896 divisible by 7? Take off the last digit (6). Double this (12) and subtract it from 89. This gives 77. Now, 77 is divisible by 7 so the original number, 896 is also divisible by 7. (With 896, it might have been just as quick to actually divide by 7!) Now try Is divisible by 7? 896, last digit 6, double 6 gives 12: Take off the last digit, 3. Double it (6) and subtract 6 from

3 Remove the 8, double it (16) and take this away from Remove the 7, double it (14) and take this away from 35, gives 21 which is divisible by 7. (PHEW!) This means that is divisible by 7. But as you can see, it may have been easier to just divide in the first place! To find the result, we DO need to divide, of course. But we know we will not get a remainder, at least! To divide, we would do: ) If the number formed by the last three digits of the number is divisible by 8, then the whole number is divisible by 8. This is not the easiest rule to apply however as you need to be able to divide the last three digits by 8 you may as well just divide the complete number by 8! BUT, if you divide the number by 2, then by 2 again, then by 2 again, (that is, you halve the number, then halve THAT number, then you halve THAT number!) you have actually divided it by 8! For example, the number from above, is NOT divisible by 8 because 806 is not divisible by 8. (Try halving, halving and halving you won t be able to halve even a second time because you get 403 on the first go.) 9 9 is similar to 3. Add the digits of the number and if that total is divisible by 9 then the number itself is also divisible by 9. Our example, , is divisible by 9 because the sum of its digits is 36, which is divisible by 9. 3

4 10 If a number ends with 0 then it is divisible by 10. For example, is divisible by 10, but is not. 11 Add the digits in the odd positions of the number; add the digits in the even positions in the number. Subtract the two sums. If the result is 0 or is divisible by 11, then the original number is divisible by 11. Example: Is divisible by 11? Add the digits in the odd positions: = 17. Add the digits in the even positions: = 6. Subtract: 17 6 = 11, which is divisible by 11. (The answer when you divide by 11 is 4894.) 12 Is the number divisible by 4? Is the number divisible by 3? If the answer to both questions is yes then the number is also divisible by 12. Example: is NOT divisible by 12 because it is not divisible by 4 (from the rule above) is NOT divisible by 12 because it is not divisible by 3 (add the digits, you will get 32). But IS divisible by 12 because it is divisible by both 3 (add the digits to get 27) and 4 (the last 2 digits, 20, make a number that is divisible by 4). That means we should get no remainder when we divide by 12: ) So, is divisible by 12 with the result of I have summarised the divisibility tests on the next page and there are some examples for you to try after that: 4

5 Divisibility tests - summary Number to divide by Rule 2 If the number is even (ends in 2, 4, 6, 8, 0), it is divisible by 2. 3 If the total of the digits is divisible by 3, then the number is divisible by 3. 4 If the last two digits of the number form a number which is divisible by 4, then the original number is divisible by 4. 5 If the number ends in 0 or 5, then it is divisible by 5. 6 If the number is even and is also divisible by 3, then it is divisible by 6. 7 Take the last digit off. Double it. Subtract it from the number remaining. Is the result divisible by 7? If yes, the original number is divisible by 7, if not, the original number is not divisible by 7. Can t tell? Repeat the process with the new result. 8 If the number formed by the last 3 digits is divisible by 8, then the original number is divisible by 8. 9 If the total of the digits is divisible by 9, then the number is divisible by If the number ends in 0 it is divisible by Add the digits in the odd positions in the number; add the digits in the even positions in the number. Subtract the two sums. If the result is 0 or is divisible by 11, then the original number is divisible by If the number is divisible by both 3 and 4, then the number is also divisible by 12. 5

6 Exercises Complete the following divisibility table for is divisible Yes Complete the following divisibility table for is Complete the following divisibility table for is divisible Yes Complete the following divisibility table for is (Solutions are on the next page) 6

7 Solutions Complete the following divisibility table for is divisible Yes Yes No No Yes Yes Complete the following divisibility table for is No No No No No Complete the following divisibility table for is divisible Yes Yes Yes Yes No Yes Complete the following divisibility table for is Yes No Yes No No No No 7