Chapter 4 of 8

All 8 chapters
  1. 1Below zero, readunderstood
  2. 2Calculating with negative numbers, readunderstood
  3. 3Multiples and factors, readunderstood
  4. 4Quick tests for dividing, readunderstood
  5. 5Prime numbers, readunderstood
  6. 6Prime factors, common factors and multiples, readunderstood
  7. 7Squares, cubes and roots, readunderstood
  8. 8Powers of ten, readunderstood

M3.1.4 Chapter 4

Quick tests for dividing

How can you tell if a number divides exactly, without doing the division?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use the last digit to test for 2, 5 and 10
  • Use the digit sum to test for 3 and 9

Ana has 4 536 beans. She wants to share them equally between 9 people. Must she do the long division to find out if it works?

A number is divisible by 5 if it divides by 5 exactly, with nothing left over. For some numbers there is a quick test. You only look at the digits.

The multiples of 10 are 10, 20, 30, 40. They all end in 0. A number ends in 0 only if it is a multiple of 10. So the test for 10 is the last digit.

The tests for 2 and 5 use the last digit too. A number is divisible by 2 if its last digit is 0, 2, 4, 6 or 8. It is divisible by 5 if its last digit is 0 or 5.

A table with the head table and last digits. The rows are 2 and 0 2 4 6 8; 5 and 0 5; 10 and 0.tablelast digits20 2 4 6 850 5100
The last digits that a multiple of 2, 5 or 10 can have.

Why only the last digit? Because 10 = 2 × 5. Every ten in a number divides by 2, by 5 and by 10. So only the ones digit can leave something over.

A table with the head number, by 2, by 5 and by 10. 340: Yes, Yes, Yes. 725: No, Yes, No. 418: Yes, No, No.numberby 2by 5by 10340YesYesYes725NoYesNo418YesNoNo
Each number is tested by its last digit. 340 ends in 0, so it passes all three tests.

Worked example

Is 2 460 divisible by 2, by 5 and by 10?

  1. Look at the last digit. It is 0.
  2. A number ending in 0 is divisible by 2, by 5 and by 10.
  3. Check: 2 460 ÷ 10 = 246, and 246 is a whole number.

Answer: Yes to all three, because the last digit is 0.

For 3 and 9 you need every digit. Add the digits up. The answer is the digit sum. If the digit sum is in the 3 times table, the number is divisible by 3. If it is in the 9 times table, the number is divisible by 9.

Draw it first

Find the digit sum of 4 536

  1. A place value chart for 4 536 with the headings thousands, hundreds, tens and ones. Step 2 adds the line 4 + 5 + 3 + 6 = 18. Step 3 adds the lines 18 = 2 × 9 and divisible by 9 and by 3.thousandshundredstensones4536

    1 Write 4 536 in a place value chart. There is one digit in each place.

  2. A place value chart for 4 536 with the headings thousands, hundreds, tens and ones. Step 2 adds the line 4 + 5 + 3 + 6 = 18. Step 3 adds the lines 18 = 2 × 9 and divisible by 9 and by 3.thousandshundredstensones45364 + 5 + 3 + 6 = 18

    2 Add the digits: 4 + 5 + 3 + 6 = 18.

  3. A place value chart for 4 536 with the headings thousands, hundreds, tens and ones. Step 2 adds the line 4 + 5 + 3 + 6 = 18. Step 3 adds the lines 18 = 2 × 9 and divisible by 9 and by 3.thousandshundredstensones45364 + 5 + 3 + 6 = 1818 = 2 × 9divisible by 9 and by 3

    3 The digit sum 18 is in the 9 times table. So 4 536 is divisible by 9, and by 3 as well.

Why does this work? Take 236. It is 2 hundreds, 3 tens and 6 ones. A hundred is 99 + 1. A ten is 9 + 1.

Base ten blocks for 236: two flats for the hundreds, three rods for the tens and six single squares for the ones, with the caption 236.236
Every flat is 99 and one more. Every rod is 9 and one more.

So 236 is 2 × 99 + 3 × 9, and then 2 + 3 + 6 more. The first part is always in the 9 times table. It is in the 3 times table too. Only the digit sum 2 + 3 + 6 = 11 can leave something over.

11 ÷ 9 leaves 2, and 236 ÷ 9 leaves 2 as well. 11 ÷ 3 leaves 2, and 236 ÷ 3 leaves 2 too. The digit sum leaves the same remainder as the whole number.

Worked example

Is 7 428 divisible by 3?

  1. Add the digits: 7 + 4 + 2 + 8 = 21.
  2. Is 21 in the 3 times table? Yes, 3 × 7 = 21.
  3. So 7 428 is divisible by 3.

Answer: Yes, because the digit sum is 21.

Worked example

Is 7 428 divisible by 9?

  1. Add the digits: 7 + 4 + 2 + 8 = 21.
  2. The 9 times table has 9 × 2 = 18 and 9 × 3 = 27.
  3. 21 is between 18 and 27, so 21 is not in the 9 times table. 21 ÷ 9 = 2 r 3.
  4. So 7 428 is not divisible by 9.

Answer: No. The digit sum is 21, and that is not divisible by nine.

See it three ways

With things

Take 27 stones. Make groups of 9. You get 3 groups and no stones are left over.

As a picture

Three groups of nine stones each, each group labelled 9 stones. Twenty seven stones in all.9 stones9 stones9 stones

In symbols

27: 2 + 7 = 9. The digit sum is 9, so 27 is divisible by 9. 27 ÷ 9 = 3.

Try it

Add the digits of the 9 times table

You need: a sheet of paper, a pencil.

  1. Write the 9 times table from 9 to 81: 9, 18, 27 and so on.
  2. Next to each number, write its digit sum.
  3. Look down the digit sums. What do you see?

What you will see: Every digit sum is 9. A number in the 9 times table has a digit sum in the 9 times table.

If you cannot try it, this is what happens.
A table with the head number and digit sum. The numbers are 9, 18, 27, 36, 45, 54, 63, 72, 81. Every digit sum is 9.numberdigit sum99189279369459549639729819

A wrong idea many people have

A number that ends in 3 is divisible by 3.

Why it feels right: The tests for 2, 5 and 10 look only at the last digit. So it seems the same should work for 3.

The test that breaks it: Try 13. 13 ÷ 3 leaves 1. Now try 33. 33 ÷ 3 = 11 with nothing left. Both end in 3, but only one divides.

What is true: For 3 and 9 every digit counts. Add them up and test the digit sum. 1 + 3 = 4 is not in the 3 times table, but 3 + 3 = 6 is.

Key words

divisible
can be divided exactly, with nothing left over.
digit sum
the answer when you add up all the digits of a number.

Back to your guess

The answer is B: A number is divisible by 5 if its last digit is 0 or 5.

Only the last digit matters for 5. A number is divisible by 5 when it ends in 0 or 5. The digit sum is for a different test, the one for 3 and 9.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which of these numbers is divisible by 5?

    Tap your answer.

    1. Yes, that is right. 1 115 ends in 5, so it is divisible by 5. 1 062 ends in 2 and 2 004 ends in 4.

    2. Not this one. You may have looked at the digit sum, 6. The digit sum is for 3 and 9. For 5, look at the last digit, which is 4. Go to M3.1.4. Try another.

    3. Not this one. This number ends in 2. The last digit must be 0 or 5 for a number to divide by 5. Look at the tests again. Go to M3.1.4. Try another.

    See the answer

    1 115

    1 115 ends in 5, so it is divisible by 5. 1 062 ends in 2 and 2 004 ends in 4.

    • If you chose B: You may have looked at the digit sum, 6. The digit sum is for 3 and 9. For 5, look at the last digit, which is 4. Go to M3.1.4.
    • If you chose C: This number ends in 2. The last digit must be 0 or 5 for a number to divide by 5. Look at the tests again. Go to M3.1.4.

  2. Which of these numbers is divisible by 10?

    Tap your answer.

    1. Not this one. This number ends in 4, so it is divisible by 2. But a number divisible by 10 must end in 0. Go to M3.1.4. Try another.

    2. Yes, that is right. 3 840 ends in 0, so it is a multiple of 10. 3 840 = 384 × 10.

    3. Not this one. This number ends in 5, so it is divisible by 5. But a number divisible by 10 must end in 0. Go to M3.1.4. Try another.

    See the answer

    3 840

    3 840 ends in 0, so it is a multiple of 10. 3 840 = 384 × 10.

    • If you chose A: This number ends in 4, so it is divisible by 2. But a number divisible by 10 must end in 0. Go to M3.1.4.
    • If you chose C: This number ends in 5, so it is divisible by 5. But a number divisible by 10 must end in 0. Go to M3.1.4.

  3. Which of these numbers is divisible by 3? Use the digit sum.

    Tap your answer.

    1. Yes, that is right. 5 + 8 + 1 + 7 = 21, and 21 = 3 × 7. So 5 817 is divisible by 3. Check: 5 817 ÷ 3 = 1 939.

    2. Not this one. This number ends in 3, but the last digit does not decide it. The digit sum is 17, which is not in the 3 times table. Go to M3.1.4. Try another.

    3. Not this one. This number ends in 0, so it is divisible by 10. For 3 you need the digit sum, and 14 is not in the 3 times table. Go to M3.1.4. Try another.

    See the answer

    5 817

    5 + 8 + 1 + 7 = 21, and 21 = 3 × 7. So 5 817 is divisible by 3. Check: 5 817 ÷ 3 = 1 939.

    • If you chose B: This number ends in 3, but the last digit does not decide it. The digit sum is 17, which is not in the 3 times table. Go to M3.1.4.
    • If you chose C: This number ends in 0, so it is divisible by 10. For 3 you need the digit sum, and 14 is not in the 3 times table. Go to M3.1.4.

  4. Which of these numbers is divisible by 9? Use the digit sum.

    Tap your answer.

    1. Not this one. The digit sum is 21. That is in the 3 times table, so 6 726 is divisible by 3. But 21 is not in the 9 times table, so it is not divisible by 9. Go to M3.1.4. Try another.

    2. Not this one. The digit sum is 16. That is not in the 9 times table, so 6 703 is not divisible by 9. Go to M3.1.4. Try another.

    3. Yes, that is right. 6 + 7 + 2 + 3 = 18, and 18 = 2 × 9. So 6 723 is divisible by 9. Check: 6 723 ÷ 9 = 747.

    See the answer

    6 723

    6 + 7 + 2 + 3 = 18, and 18 = 2 × 9. So 6 723 is divisible by 9. Check: 6 723 ÷ 9 = 747.

    • If you chose A: The digit sum is 21. That is in the 3 times table, so 6 726 is divisible by 3. But 21 is not in the 9 times table, so it is not divisible by 9. Go to M3.1.4.
    • If you chose B: The digit sum is 16. That is not in the 9 times table, so 6 703 is not divisible by 9. Go to M3.1.4.

  5. Which of these numbers is divisible by both 2 and 3?

    Tap your answer.

    1. Not this one. The digit sum is 15, so it is divisible by 3. But it ends in 3, so it is not divisible by 2. Go to M3.1.4. Try another.

    2. Yes, that is right. 4 536 ends in 6, so 2 divides it. Its digit sum is 4 + 5 + 3 + 6 = 18, so 3 divides it too.

    3. Not this one. This number ends in 8, so it is divisible by 2. But its digit sum is 20, which is not in the 3 times table. Go to M3.1.4. Try another.

    See the answer

    4 536

    4 536 ends in 6, so 2 divides it. Its digit sum is 4 + 5 + 3 + 6 = 18, so 3 divides it too.

    • If you chose A: The digit sum is 15, so it is divisible by 3. But it ends in 3, so it is not divisible by 2. Go to M3.1.4.
    • If you chose C: This number ends in 8, so it is divisible by 2. But its digit sum is 20, which is not in the 3 times table. Go to M3.1.4.

  6. What is the digit sum of 5 817?

    See the answer

    21

    5 + 8 + 1 + 7 = 21.

  7. Amina writes 52, then one more digit on the end. Which last digit makes her three digit number divisible by 9?

    See the answer

    2

    5 + 2 = 7. The next number in the 9 times table is 9, and 7 + 2 = 9. So the digit is 2, and 522 = 9 × 58.

  8. From an earlier chapter

    List all the factors of 18.

    See the answer

    1, 2, 3, 6, 9 and 18

    The factor pairs of 18 are 1 × 18, 2 × 9 and 3 × 6. Write the numbers in order.

Keep it

Copy this into your exercise book.

Ends in 0, 2, 4, 6 or 8: divisible by 2. Ends in 0 or 5: by 5. Ends in 0: by 10. Digit sum in the 3 or 9 table: the number is too.

Draw: A number such as 4 536 in four boxes, with its digits added up underneath. Write 4 + 5 + 3 + 6 = 18.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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