divisibility tests year 7 mathematics.pptx

divisibility tests chapter 1

1. Tests for divisibility ÷?

2. Is 42894 divisible by 3? 4+2+8+9+4=27 2+7=9 9 is a multiple of 3, so 42894 is as well! Add digits And again Is 830276 divisible by 3? 8+3+0+2+7+6=25 2+5=7 7 isn’t a multiple of 3, so 830276 isn’t either Add digits And again We will prove why this method works at the end of the lesson! What other divisibility tests do you know?

3. Test 2 3 4 5 6 8 9 Even Digits sum to a multiple of 3 Last 2 digits are divisible by 4 Ends in 5 or 0 Divisible by 2 and 3 Can be halved 3 times Digits sum to a multiple of 9 Tests for divisibility Divisible by? 2 3 4 5 6 8 9 38 96 80 225 174 416 360 1. Complete the descriptions: 2. Tick or cross for divisibility by each number

4. Divisible by 2 Divisible by 3 Divisible by 5 14 70 51 25 98 831 110 360 What do you notice about the numbers in the parts where the circles overlap? 3. Using the divisibility tests, place each number in the Venn Diagram 24 342 45 75 65 150 14 75 12 24 98 125 25 110 30 45 150 100 51 342 256 65 360 123 70 831 225 225 125 30 100 256 123 12 Multiples of 10 Multiples of 6 Multiples of 15 Multiples of 30

5. Is 42894 divisible by 3? 4+2+8+9+4=27 2+7=9 9 is a multiple of 3, so 42894 is as well! Add digits And again Proof for 2 digit numbers Represent the number algebraically as 10a+b 10a+b = 9a+( a+b ) divisible by 3 If this is divisible by 3, then so is the whole number! Proof for 3 digit numbers Represent the number algebraically as 100a+10b+c 100a+10b+c = 99a+9b+( a+b+c ) divisible by 3 If this is divisible by 3, then so is the whole number!

6. Divisibility by 7 Double the units and subtract this from the truncated number Eg 273 Eg 632 There are similar tests for divisibility by 11, 13, 17 and 19. Complete the following table by applying the tests described. The first one has been done as an example. Eg 60473 If the test gives a large number, just keep applying it till you get a smaller one! 21 is a multiple of 7 so 273 is as well If this gives a multiple of 7 then the original number is as well! Truncate the number by leaving off the units 273 → 27 – 6 = 21 59 isn’t a multiple of 7 so 273 isn’t either 632 → 63 – 4 = 59 60473 → 6047 – 6 = 6041 6041 → 604 – 2 = 602 602 → 60 – 4 = 56 56 is a multiple of 7 so 60473 is as well

7. Divisibility by What to do to truncated number Example 11 - units Is 29174 divisible by 11? 29174 → 2917-4-2913 2913 → 291-3=288 288 → 28-8=20 20 isn’t divisible by 11 so 19151 isn’t either 13 + unitsx4 Is 50661 divisible by 13? 17 - unitsx5 Is 67252 divisible by 17? 19 + unitsx2 Is 40538 divisible by 19?

8. Divisibility by What to do to truncated number Example 11 - units Is 29174 divisible by 11? 29174 → 2917-4-2913 2913 → 291-3=288 288 → 28-8=20 20 isn’t divisible by 11 so 19151 isn’t either 13 + unitsx4 I s 50661 divisible by 13? 50661 → 5066+4=5070 5070 → 507+0=507 507 → 50+28=78 78 is divisible by 13 so 50661 is as well 17 - unitsx5 Is 67252 divisible by 17? 67252 → 6725-10=6715 6715 → 671-25=646 646 → 64-30=34 34 is divisible by 17 so 67252 is as well 19 + unitsx2 Is 40538 divisible by 19? 40538 → 4053+16=4069 4069 → 406+18=424 424 → 42+8=50 50 isn’t divisible by 19 so 40538 isn’t either

9. A proof of why this works for 2 digit numbers Let the 2 digit number be The test would give the value If the original number is divisible by 7, then for some integer a Rearranging this, Substitute this into the value the test gives Which is divisible by 7 Hence the test works! Divisibility by 7 Double the units and subtract this from the truncated number If this gives a multiple of 7 then the original number is as well! Truncate the number by leaving off the units Can you construct similar proofs for divisibility by 11, 13, 17 and 19?

10. Divisibility by 13 Let the 2 digit number be The test would give the value If the original number is divisible by 13, then for some integer a Rearranging this, Substitute this into the value the test gives Which is divisible by 13 Hence the test works! Divisibility by 11 Let the 2 digit number be The test would give the value If the original number is divisible by 11, then for some integer a Rearranging this, Substitute this into the value the test gives Which is divisible by 11 Hence the test works!

11. Divisibility by 19 Let the 2 digit number be The test would give the value If the original number is divisible by 19, then for some integer a Rearranging this, Substitute this into the value the test gives Which is divisible by 19 Hence the test works! Divisibility by 17 Let the 2 digit number be The test would give the value If the original number is divisible by 17, then for some integer a Rearranging this, Substitute this into the value the test gives Which is divisible by 17 Hence the test works!

12. How many different solutions can you find, without a calculator, using the digits 1 to 9 once only in this sum? 98765432 isn’t divisible by 3 (and hence 6 or 9) 98765432 isn’t divisible by 5 Leaving 2,4,7 and 8 to test... leaving 98765432 ÷ 8 = 12345679 98765432 ÷ 2 = 49382716 = 98765432 98765432 ÷ 7 = not a whole number 98765432 ÷ 4 = 24691358 2 appears twice (and 5 not at all) 4 appears twice (and 7 not at all) 98765432 x 1 = 98765432 obviously! To investigate other possibilities, it is easier to consider 98765432

13. Only one choice of the digit d gives prime numbers when you read across and down in the diagram below. Which digit is d? C 7 5 d 7 1 3 B 6 A 5 Hint: consider the test for divisibility by 3 Vertically, 5 + 7 = 12 so d cannot be 6 Horizontally, 1 + 3 = 4 so d cannot be 5 Which leaves d = 7

14. 15 108 224 144 8 315 1 2 3 4 5 6 7 8 9 Task A : The boxes at the end of each row and the foot of each column give the result of multiplying the three numbers in that row or column. Can you arrange the numbers 1 to 9 in the grid? Problem solving with divisibility tests

15. Smallest __________ 5432 534 Number must be even Digits must sum to a multiple of 3 Don’t use the 2 as this is the smallest value you can remove to obtain a multiple of 3 Must be a multiple of 2 and 3 45768 87564 Last 2 digits must be divisible by 4 Multiple of 2 Task B : What is the largest multiple you can make using the digits below? You don’t have to use each digit and can use each one at most once Multiple of 3 Multiple of 6 2 3 4 5 Task C : What are the smallest and largest multiples of 4 you can make using all the digits below? 4 5 6 7 8 Largest ___________ 543

16. 123654 321654 2 nd , 4 th and 6 th digits must be even, so 1 st , 3 rd and 5 th are odd 5 th digit must be 5 first 3 digits must sum to a multiple of 3 testing possible combinations with these restrictions yields two solutions: even odd 3,608,528,850,368,400,786,036,725 has 25 digits and divides by 25. BUT... it you just take the first  n  digits, the result will divide by  n . For example, 360852 are the first 6 digits and 360852 divides by 6. Task D : Using the digits 1 to 6, create a 6 digit number so that the first two digits are divisible by two, the first three digits are divisible by three, etc How many answers are there? Task E : Do the same using the digits 0 to 9 (one answer!)

17. Test 2 3 4 5 6 8 9 Tests for divisibility Divisible by? 2 3 4 5 6 8 9 38 96 80 225 174 416 360 1. Complete the descriptions: 2. Tick or cross for divisibility by each number

18. Divisible by 2 Divisible by 3 Divisible by 5 What do you notice about the numbers in the parts where the circles overlap? 3. Using the divisibility tests, place each number in the Venn Diagram 14 75 12 24 98 125 25 110 30 45 150 100 51 342 256 65 360 123 70 831 225

19. Divisibility by What to do to truncated number Example 11 - units Is 29174 divisible by 11? 29174 → 2917-4-2913 2913 → 291-3=288 288 → 28-8=20 20 isn’t divisible by 11 so 19151 isn’t either 13 + unitsx4 Is 50661 divisible by 13? 17 - unitsx5 Is 67252 divisible by 17? 19 + unitsx2 Is 40538 divisible by 19?

20. 15 108 224 144 8 315 Task A : The boxes at the end of each row and the foot of each column give the result of multiplying the three numbers in that row or column. Can you arrange the numbers 1 to 9 in the grid? Problem solving with divisibility tests

21. Largest ___________ Smallest __________ Multiple of 2 Task B : What is the largest multiple you can make using the digits below? You don’t have to use each digit and can use each one at most once Multiple of 3 Multiple of 6 2 3 4 5 Task C : What are the smallest and largest multiples of 4 you can make using all the digits below? 4 5 6 7 8

22. 3,608,528,850,368,400,786,036,725 has 25 digits and divides by 25. BUT... it you just take the first  n  digits, the result will divide by  n . For example, 360852 are the first 6 digits and 360852 divides by 6. Task D : Using the digits 1 to 6, create a 6 digit number so that the first two digits are divisible by two, the first three digits are divisible by three, etc How many answers are there? Task E : Do the same using the digits 0 to 9 (one answer!)