◄UNIT 2:Addition and subtraction of integersUNIT 4:Equivalent fractions and operations►

UNIT 3:Prime factorisation and divisibility tests

  • UNIT 3:Prime factorisation and divisibility tests

    3.1 Prime factorisation of numbers and its uniqueness

    Activity 3.1
    Prime factorise the following numbers. Explain the steps to your answer.

    (a) 60                                   (b) 180

    Tip:
    A prime number is a number that has only two different factors. That is 1
    and itself.
    Some examples of prime numbers are 2, 3, 5, 7, 11 and 13. We can write a
    number using its prime factors.
    Look at the following:

    Prime factorise 40.

    m 40 = 2 × 2 × 2 × 5

    Example 3.1
    Prime factorise 30

    Solution

    s 30 = 2 × 3 × 5

    Practice Activity 3.1
    Write each of the following as product of its prime factors.
    1. 40       2. 120       3. 170       4. 80          5. 200
    Prime factorise the following numbers. Explain your answer.
    6. 320     7. 540       8. 670       9. 560       10. 132

    11. 366     12. 144   13. 266    14. 470      15. 920

    3.2 Using indices as shorthand for repeated factors
    Activity 3.2
    Factorise the following numbers. Use indices (or powers) to show repeated
    prime factors.
    (a) Prime factorise 120.

    (b) Prime factorise 280. Explain what you have noticed.

    Tip:
    We can express numbers as products of prime factors. We can use powers
    or indices on repeated prime factors. For example, prime factorise.
    (a) 68                              (b) 16.

    Express them using indices.

    a)
    s

     68 = 2 × 2 × 17
    = 22 × 17 (This is because 2 × 2
    is such that 2 is repeated two

    times)



    ,

    16 = 2 × 2 × 2 × 2
    = 24 (This is because 2 is repeated
    4 times)

    Example 3.2
    Prime factorise 60. Show the prime factors using indices.

    Solution 
    m


    60 = 2 × 2 ×3 ×5
    60 = 22 × 3 × 5

    22 is 2 to the power 2 or 2 × 2 (2 two times)

    Therefore 60 = 22 × 3 × 5 has been written using factors in powers/indices.
    Now prime factorise 72. Show the prime factors using indices. Explain

    the steps you followed to arrive at your answer. Present your findings

    Practice Activity 3.2
    Prime factorise the following numbers. Show their prime factors using
    indices (or powers).
    1. 27          2. 75          3. 36                4. 76
    Prime factorise the following. Express prime factors in indices form and
    explain.
    5. 98         6. 48          7. 25                 8. 64
    9. 45        10. 106       11. 54              12. 74

    3.3 Calculation of the Least Common Multiple (LCM)
    What is the multiple of a number? When you have two numbers, like 5
    and 6, you can list their multiples. There will be a common multiple. The
    smallest of the common multiples is the Least Common Multiple.

    Now, do the following activity.

    Activity 3.3
    Find the Least Common Multiple of;
    (a) 3, 9 and 12            (b) 3 , 6 and 9
    (c) 3, 4 and 8              (d) 4, 5, and 8
    Find some examples where you can apply the LCM to daily life. Discuss

    your findings

    m

    Practice Activity 3.3
    1. Find the LCM of the following numbers.
    (a) 2, 5 and 10    (b) 5, 6 and 9      (c) 2, 6 and 8
    2. Find the LCM of the numbers below. Present your answers.
    (a) 6, 15 and 20    (b) 4, 5 and 10   (c) 3, 4 and 5
    3. Find the LCM of the numbers below. Explain the steps to your answer.
    (a) 4, 5 and 12      (b) 4, 6 and 9      (c) 6, 15 and 10
    (d) 12, 18              (e) 10, 15, 9

    3.4 Calculation of Greatest Common Factors (GCF)
    What is the GCF of 18, 12 and 24? Start dividing with the smallest prime
    factor that divides all the numbers. Continue dividing until there is no other
    prime factor that can divide all the numbers.
    m

    Hint:
    There is no common divisor for 3, 2, 12. So
    we stop division






    Therefore the GCF of 12, 18 and 24 is:
    = 2 × 3
    = 6

    Activity 3.4
    Find the Greatest Common Factor (GCF) of the following numbers.
    (a) 36 and 39        (b) 42 and 48
    (c) 9, 18 and 27    (d) 15, 30 and 35
    Explain the steps to your answer.

    Discuss daily life examples where you use the GCF.

    Example 3.4
    Find the Greatest Common Factor (GCF) of 28, 42 and 56.

    Solution
    Method 1
    Start dividing by the smallest                                  Method 2
    prime number that divides                                Express 28, 42 and 56 in indices forms: 
    all the numbers.                                                28 = 2 × 2 × 7 = 22 × 7 
    m        42 = 2 × 3 × 7
             56 = 2 × 2 × 2 × 7 = 23 × 7
             Observation on Common factors in
              indices:
             2, 22, 23 and 7 are common. 3 is not
             common.
              So, GCF is 2 × 7 = 14.
    Therefore GCF is 2 × 7 = 14                        We use common factors with lowest indices.

    Practice Activity 3.4
    Find the Greatest Common Factor (GCF) of the numbers below.
    1. 14, 20 and 36        2. 24, 36 and 40       3. 72, 84 and 108
    4. 84, 140 and 224    5. 42, 70 and 112      6. 220 and 360
    Calculate the GCF of the following. Discuss your steps.
    7. 54 and 90            8. 45, 60 and 750        9. 250, 450 and 750
    10. 180, 360 and 630
    3.5 Divisibility test for 2
    Activity 3.5
    Divide the following numbers by 2.
    (a) 3 241     (b) 573 428     (c) 361 800      (d) 520 042
    • Which numbers are divisible by 2? Check their last digits. What do
    you notice?
    • Which numbers are not divisible by 2? Check their last digits. What
    do you notice?

    • What can you say about the last digit of the numbers divisible by 2?

    Present your findings.
    Tip:
    A number is divisible by 2 if the last digit is an even number or zero.
    Example 3.5
    1. Is 49 140 divisible by 2?
    Solution
    The last digit in 49 140 is 0.
    Therefore the number 49 140 is divisible by 2.
    2. Test if the following are divisible by 2.
      (a) 90 712                              (b) 90 721

    Solution
    (a) The last digit 2 in 90 712 is an even number.
    Therefore the number 90 712 is divisible by 2.
    (b) The last digit 1 is an odd even number.
    Therefore, 90 721 is not divisible by 2.
    Practice Activity 3.5
    Which of the following numbers are divisible by 2?
    1. 4 480          2. 6 429         3. 5 258
    4. 21 224        5. 49 242       6. 15 504
    7. 470 881      8. 636 027     9. 36 085
    Test and write numbers divisible by 2. Discuss how you found your answers.
    10. 52 100      11. 148 516   12. 462 946
    13. 90 712      14. 54 213     15. 41 768
    16. 87 742      17. 49 112      18. 214 332

    3.6 Divisibility test for 3
    Activity 3.6
    • Divide the following numbers by 3.
    (a) 39        (b) 214       (c) 171         (d) 8 811
    Find the sum of the digits of each number above. Divide the sum for each
    number by 3. What do you discover?
    Present your findings.
    Tip:

    A number is divisible by 3 if the sum of its digits is a multiple of 3.

    Example 3.6
    (a) Test if 1 824 is divisible by 3?

    (b) Test if 23 416 is divisible by 3.

    Solution
    (a) • Add the digits for the number 1 824.
    1 + 8 + 2 + 4 = 15. Now, 15 ÷ 3 = 5. So 15 is divisible by 3.
    Therefore, 1 824 is divisible by 3.

    (b) • Add the digits for 23 416. We have: 2 + 3 + 4 + 1 + 6 = 16. Now
    16 ÷ 3 = 5 with remainder of 1.
    So, 16 is not divisible by 3.

    Therefore, 23 416 is not divisible by 3.

    Practice Activity 3.6
    Test and give the numbers that are divisible by 3. Explain the steps to your
    answers.
    1. 1 836           2. 5 613           3. 9 786
    4. 6 123           5. 56 004         6. 23 112
    7. 62 172         8. 456 312       9. 214 701
    10. 306 171    11. 178 123     12. 363 114
    13. 100 456    14. 690 390     15. 120 300

    3.7 Divisibility test for 4
    Activity 3.7
    • Divide the following numbers by 4.
    (a) 2 472       (b) 2 814        (c) 17 936
    Which of them are divisible by 4?
    • Test whether the last 2 digits of each number is divisible by 4 or not.
    What do you notice?

    Present your findings.

    Tip:
    A number is divisible by 4 if the last 2 digits form a number divisible by 4.

    Example 3.7
    (a) Is 456 312 divisible by 4?

    (b) Is 106 526 divisible by 4?

    Solution
    (a) The last 2 digits of 456 312 forms 12. Now, 12 ÷ 4 = 3.
    So, 12 is divisible by 4.
    Therefore, 456 312 is divisible by 4.
    (b) The last 2 digits of 106 526 forms 26. Now, 26 ÷4 = 6 with remainder
    of 2. So, 26 is not divisible by 4.

    Therefore, 106 526 is not divisible by 4.

    Practice Activity 3.7
    Test which of these numbers are divisible by 4.
    1. 839 016         2. 7 936              3. 49 424
    4. 873 008         5. 990 004          6. 182 510
    7. 52 850           8. 91 044            9. 41 928
    Test for numbers divisible by 4. Discuss your steps.
    10. 3 148          11. 98 541         12. 83 710
    13. 426 940      14. 201 084       15. 390 712

    3.8 Divisibility test for 5
    Activity 3.8
    Divide the following numbers by 5.
    (a) 99 000      (b) 27 435      (c) 47 861        (d) 78 390
    Which numbers are divisible by 5? Check their last digit.
    Which numbers are not divisible by 5? Check their last digit.
    What do you notice about the last digit of numbers divisible by 5?
    Discuss your findings.
    Tip:

    A number is divisible by 5 if its last digit is 0 or 5.

    Example 3.8
    Which of the following numbers is divisible by 5?
    (a) 56 480        (b) 225 445           (c) 741 024
    Solution
    (a) 56 480 has the last digit 0. Therefore, 56 480 is divisible by 5.
    (b) 225 445 has the last digit 5. Therefore, 225 445 is divisible by 5.
    (c) 741 024 has the last digit 4. Therefore, 741 024 is not divisible by 5.
    Practice Activity 3.8
    Test to find the numbers are divisible by 5.
    1. 487 200 2. 578 425 3. 140 265
    4. 859 420 5. 718 426 6. 419 347

    Test for numbers divisible by 5. Explain your steps.
    7. 736 920 8. 878 945 9. 572 315
    10. 640 635 11. 670 670 12. 654 285
    13. 563 759 14. 410 458 15. 369 000
    3.9 Divisibility test for 6
    Activity 3.9
    Look at the numbers below.
    (a) 336     (b) 690      (c) 4 878     (d) 194     (e) 736
    Divide the numbers by 2.
    Divide the numbers by 3 again.
    Divide the same numbers by 6.
    What do you notice about the numbers?
    Discuss your findings.

    Tip:

    A number is divisible by 6 if it is also divisible by 2 and 3.

    Example 3.9
    Which of the numbers below is divisible by 6? Explain your steps.
    (a) 2 700                                          (b) 458 716
    Solution
    (a) • The last digit for 2 700 is 0. So 2 700 is divisible by 2.
    • 2 + 7 + 0 + 0 = 9. The sum of the digits of 2 700 is 9. So 9 is
    divisible by 3. Therefore, 2 700 is divisible by 3.
    • Finally, 2 700 is divisible by 6.
    (b) • The last digit of 458 716 is 6. Now, 6 is an even number. Thus,
    458 716 divisible by 2.
    • 4 + 5 + 8 + 7 + 1 + 6 = 31. The sum of the digits of 458 716 is 31.
    Now, 31 ÷ 3 = 10 rem 1, or 31 is not divisible by 3. Thus, 458 716
    is not divisible by 3.
    • Finally, 458 716 is not divisible by 6.
    Practice Activity 3.9
    Test and give numbers that are divisible by 6.
    1. 70 032           2. 54 451            3. 46 008
    4. 82 092           5. 14 256            6. 85 728
    Test to find numbers divisible by 6. Discuss your steps.
    7. 458 710        8. 51 200             9. 216
    10. 144            11. 928                12. 93 621
    13. 3 759         14. 48 780           15. 56 800
    3.10 Divisibility test for 8
    Activity 3.10
    • Divide the numbers below by 8.
    (a) 5 328     (b) 17 428      (c) 93 640
    • Now form a number from the last three digits of each number. Divide

    your number by 8. What do you notice? Explain your observations.

    Tip:
    A number is divisible by 8 if the last three digits form a number divisible
    by 8.
    Example 3.10
    Investigate for the numbers that are divisible by 8.
    (a) 404 320      (b) 200 072            (c) 323 638
    Solution
    Check if the number formed by the last 3 digits is divisible by 8.
    (a) From 404 320, the last digits form 320. Now 320 ÷ 8 = 40. Since 320
    is divisible by 8, thus 404 320 is divisible by 8.
    (b) From 202 072, the last 3 digits form 072. Now 072 ÷ 8 = 9. Since 072
    is divisible by 8, thus 202 072 is divisible by 8.
    (c) From 323 638, the last 3 digits form 638. Now 638 ÷ 8 = 79 with
    remainder of 6, is not divisible by 8. Thus, 323 638 is not divisible by 8.
    Practice Activity 3.10
    Test and give the numbers that are divisible by 8.
    1. 842 056          2. 300 400         3. 642 323
    4. 374 816          5. 322 642         6. 138 648
    7. 183 257          8. 768 265         9. 543 120
    Test and write the numbers that are divisible by 8. Explain your steps.
    10. 679 168       11. 217 800        12. 436 756
    13. 374 912       14. 276 480        15. 248 263

    3.11 Divisibility Test for 9

    Activity 3.11
    • Divide these numbers by 9.
    (a) 8 109         (b) 2 916       (c) 20 007       (d) 108 450
    • Add the digits of the numbers given above.
    Divide the sum of the digits by 9. Are they all divisible by 9?
    • What do you notice about numbers divisible by 9? Present your
    findings.

    Tip:
    A number is divisible by 9 if the sum of its digits form a number divisible
    by 9.
    Example 3.11
    Which of the following numbers is divisible by 9?
    (a) 64 737 (b) 607 131 (c) 128 000
    Solution
    Step 1: Add the digits of the numbers.
    (a) 64 737 : 6 + 4 + 7 + 3 + 7 = 27
    (b) 607 131 : 6 + 0 + 7 + 1 + 3 + 1 = 18
    (c) 128 000 : 1+ 2+ 8+ 0 + 0 + 0 = 11
    Step 2: Divide the sum by 9. State which numbers are divisible by 9.
    (a) 27 ÷ 9 = 3. Therefore 64 737 is divisible by 9.
    (b) 18 ÷ 9 = 2. Therefore 607 131 is divisible by 9.
    (c) 11 ÷ 9 = 1 with remainder of 2. Therefore 128 000 is not
      divisible by 9.
    Practice Activity 3.11
    1. Test and write the numbers that are divisible by 9.
    (a) 98 541        (b) 49 041        (c) 903 132
    (d) 383 121      (e) 394 020      (f) 42 568
    (g) 34 679        heart 721 800      (i) 530 280
    2. Test and write the numbers divisible by 9. Discuss your answer.
    (a) 713 610      (b) 819 234      (c) 999 045

    (d) 515 230       (e) 304 133

    3.12 Divisibility test for 10
    Activity 3.12
    Divide the following numbers by 10.
    (a) 8 730     (b) 6 940      (c) 5 285     (d) 94 000      (e) 20 184
    Which numbers are divisible by 10?
    Which numbers are not divisible by 10?
    Check the numbers that are not divisible by 10 again? What are their
    last digits? Discuss your observations.

    Tip:
    A number is divisible by 10 if it ends with 0.

    Example 3.12
    Which of the following numbers are divisible by 10?
    (a) 49 140    (b) 199 000      (c) 447 861      (d) 872 930
    Solution
    The numbers with a last digit of 0 are:
    (a) 49 140      (b) 199 000 and      (d) 872 930
    Therefore 49 140, 199 000, 872 930 are divisible by 10.
    (c) 447 861 is not divisible by 10. It ends with 1.
    Practice Activity 3.12
    1. Which of the following numbers are divisible by 10?
    (a) 1 000 000      (b) 405 330        (c) 555 355
    (d) 725 660          (e) 554 740
    2. Test which numbers are divisible by 10. Discuss and present your findings. 
    (a) 874 930         (b) 582 140         (c) 529 900

    (d) 81 420           (e) 793 004

    3. List five numbers that are divisible by 10.n
    4. Workers offloaded a lorry with 50 000
    books. The books are to be shared by 10


    3.13 Divisibility test for 11
    Look at 2 463. The digits 4, 3 are alternate. Similarly, 2 and 6 are alternate
    digits. Let us do the activity below.
    Activity 3.13
    • Get the sums of the alternate digits in each of the following. Then
    find their differences.
    (a) 3 190    (b) 3 465     (c) 2 376      (d) 18 931
    Divide each of the numbers by 11. Check the difference of alternate
    digits for those numbers divisible by 11. Present your findings

    Tip: If the difference of the sums of alternate digits is 0, 11 or a multiple
    of 11, then the number is divisible by 11.•

     Example 3.13
    1. Is 23 760 divisible by 11?
    Solution
    • Add alternate digits: (2 + 7 + 0) = 9 and (3 + 6) = 9
    • Find their difference 9 – 9 = 0
    Difference is 0. Therefore, 23 760 is divisible by 11.
    2. Is 934 010 divisible by 11?
    Solution
    • Add alternate digits: (9 + 4 + 1) = 14
    (3 + 0 + 0) = 3
    Find their difference
    14 – 3 = 11
    Difference is 11. Therefore, 934 010 is divisible by 11.
    3. Is 575 814 divisible by 11?
    Solution
    • Add alternate digits: (5 + 5 + 1) = 11
    (7 + 8 + 4) = 19
    • Find their difference
    19 – 11 = 8. The difference 8 is not divisible by 8.
    Therefore, 575 814 is not divisible by 11. 

    Practice Activity 3.13
    1. Test which of the numbers below is divisible by 11?
    (a) 469 246                  (b) 329 856
    (c) 986 832                  (d) 912 857
    2. Write the numbers that are divisible by 11. Discuss your test.
    (a) 102 762                  (b) 105 820
    (c) 862 211                   (d) 422 939
    3. Test for numbers divisible by 11 from below. Explain your steps.
    (a) 352 274                  (b) 329 835
    (c) 422 940                   (d) 9 625

    3.14 Divisibility test for 12
    Activity 3.14
    • Divide the following numbers by 12.
    (a) 1 524 (b) 1 320 (c) 3 936 (d) 2 544 (e) 5 076
    Divide each number by 3. Divide each of the numbers by 4.
    Are all the numbers divisible by 12, 3 and 4?
    • Which numbers are divisible by both 3 and 4? Which numbers are divisible by 12? Discuss your findings.
    Tip:
    A number is divisible by 12 if it is divisible by both 3 and 4.
    Example 3.14
    Test and state the number that is divisible by 12.
    182 844 or 644 346
    Solution
    Hint: Carry out divisibility tests for both 3 and 4.
    Step 1: Divisibility test for 3. Add the digits and divide by 3.
    • 182 844: 1 + 8 + 2 + 8 + 4 + 4 = 27. Now 27 ÷ 3 = 9.
    Thus, 182 844 is divisible by 3.
    • 644 346: 6 + 4 + 4 + 3 + 4 + 6 = 27. Now 27 ÷ 3 = 9.
    Thus, 644 346 is divisible by 3.
    Step 2: Divisibility test for 4. Divide the number formed by the last 2
    digits of each number by 4.
    • From 182 844; we have 44 ÷ 4 = 11. So 182 844 is divisible
    by 4.
    • From 644 346; 46 ÷ 4 = 11 with remainder of 2. Therefore, 46
    is not divisible by 4.
    So 644 346 is not divisible by 4.
    Step 3: Conclusion – number divisible by 12.
    • From Steps 1 and 2, 182 844 is divisible by both 3 and 4.
    Therefore, 182 844 is divisible by 12.
    • From Steps 1 and 2, 644 346 is divisible by 3 and not by 4.
    Thus, 644 346 is not divisible by 12.

    Practice Activity 3.14
    Find the numbers that are divisible by 12.
    1. 3 360         2. 2 724          3. 9 684
    4. 8 676         5. 89 184        6. 58 968
    7. 39 300       8. 26 716        9. 541 656
    Test and write the numbers divisible by 12. Discuss your steps.
    10. 933 216   11. 753 072     12. 665 580
    13. 582 100   14. 403 560

    Revision Activity 3
    1. Prime factorise the numbers below using indices.
    (a) 240        (b) 300          (c) 1 000
    2. Find the Least Common Multiple of the following.
    (a) 6, 9 and 12                  (b) 4, 8 and 10
    (c) 8, 10 and 1                  (d) 10, 12 and 15
    3. Find the Greatest Common Factor of the following. Explain your
    answer.
    (a) 48, 40 and 72             (b) 100, 120 and 150
    4. Identify the numbers divisible by 2 below.
    (a) 649 426      (b) 241 233        (c) 792 400
    5. Which of the following numbers are divisible by 3?
    (a) 300 012     (b) 400 560         (c) 450 106
    6. Name the numbers that are divisible by 4. Present your answers.
    (a) 480 120     (b) 820 440         (c) 541 610
    7. Which numbers are divisible by 5?
    (a) 400 255     (b) 426 451         (c) 728 400
    8. Identify the numbers that are divisible by 6. Explain your answers.
    (a) 403 560     (b) 67 260           (c) 2 724
    9. Name the numbers divisible by 8. Discuss your steps.
    (a) 868 562     (b) 480 240         (c) 976 861
    10. Which of the numbers below is divisible by 9?
    (a) 810 720     (b) 820 503          (c) 413 333
    11. Which of the numbers is divisible by 10?
    (a) 716 300     (b) 633 420           (c) 660 855
    12. Name the numbers divisible by 11. Explain your answers.
    (a) 467 181     (b) 891 484           (c) 541 656
    13. Which of the following numbers are divisible by 12? Discuss your
    steps.
    (a) 891 480     (b) 556 680           (c) 497 185

    Word lis    t
    Prime factorisation                              Divisible
    Prime numbers                                    Indices (powers)
    Least Common Multiple                      Greatest Common Factor
    Divisibility tests                                    Natural numbers

    Task
    Do the following.
    (i) Read each word aloud to your friend.
    (ii) Write the meaning of each of the words above. Discuss with your friend.
    (iii) Write sentences using each of the words above. Read with your friend.

    ◄UNIT 2:Addition and subtraction of integersUNIT 4:Equivalent fractions and operations►