Mathly

Clear maths education for South African students, Grade 4 to Grade 12.

Support

  • Contact Us
  • FAQs
  • About

Account

  • Subscription Plans

Earn Free Access

  • How to get Mathly free

© 2026 Mathly. All rights reserved.

Terms & ConditionsPrivacy Policy
←
Home/Grade 6/Whole Numbers

Whole Numbers

Free

Grade 6

#

1. Place Value up to 9-Digit Numbers

In Grade 6 we work with whole numbers up to 999 999 999 — that is nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine. Every digit in a number has a place value. Starting from the right the places are: units, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions and hundred millions. We can write numbers in expanded notation to show the value of each digit separately.

Colour key:hundred millions / hundredsten millionsmillionshundred thousandsten thousands / tensthousands / units

Place value positions

Hundred
Millions		Ten
Millions		MillionsHundred
Thousands		Ten
Thousands		ThousandsHundredsTensUnits
100 000 00010 000 0001 000 000100 00010 0001 000100101

Expanded notation

To write a number in expanded notation, write the value of each non-zero digit separately and add them together:

325 407 819 = 300 000 000 + 20 000 000 + 5 000 000 + 400 000 + 7 000 + 800 + 10 + 9

Remember: zeros in expanded notation

Digits with a value of zero are not written in expanded notation because adding 0 does not change the total. For example, the 0 in the ten thousands place of 325 407 819 contributes 0 × 10 000 = 0, so it is left out.

Worked Examples

  1. 1
    Starting from the right — 6 is in the units place = 6.
  2. 2
    5 is in the tens place = 50.
  3. 3
    3 is in the hundreds place = 300.
  4. 4
    8 is in the thousands place = 8 000.
  5. 5
    2 is in the ten thousands place = 20 000.
  6. 6
    7 is in the hundred thousands place = 700 000.
  7. 7
    4 is in the millions place = 4 000 000.
Answer4 millions, 7 hundred thousands, 2 ten thousands, 8 thousands, 3 hundreds, 5 tens, 6 units
  1. 1
    300 000 000 + 20 000 000 + 5 000 000 + 400 000 + 7 000 + 800 + 10 + 9
  2. 2
    Each digit is written as its full value and added together. The 0 in the ten thousands place contributes 0 and is not included.
Answer300 000 000 + 20 000 000 + 5 000 000 + 400 000 + 7 000 + 800 + 10 + 9
  1. 1
    Find the digit 7 in the number 274 863 109.
  2. 2
    It is in the ten millions place.
  3. 3
    The value of 7 in this number is 7 × 10 000 000 = 70 000 000.
AnswerThe value of 7 in 274 863 109 is 70 000 000
⇄

2. Properties of Whole Numbers

Whole numbers follow rules — called properties — that always work, no matter which numbers you use. Knowing these properties helps you check your working and calculate mentally much faster, by reordering, regrouping or splitting numbers into friendlier parts before you add or multiply.

Colour key:commutative propertyassociative propertydistributive propertyidentity elements

The properties of whole numbers

Commutative property

a + b = b + a  and  a × b = b × a. You can add or multiply numbers in any order and get the same answer.

Associative property

(a + b) + c = a + (b + c)  and  (a × b) × c = a × (b × c). You can group numbers differently and get the same answer.

Distributive property

a × (b + c) = (a × b) + (a × c). Multiplying a sum by a number gives the same answer as multiplying each part separately and then adding.

Identity elements

a + 0 = a (0 is the additive identity). a × 1 = a (1 is the multiplicative identity). Adding 0 or multiplying by 1 never changes a number.

Using properties to calculate mentally

1

Reorder — use the commutative property to bring numbers that combine easily (like a pair that adds to 10, 100 or 1 000) next to each other.

2

Regroup — use the associative property to combine the friendly pair first, before dealing with the rest.

3

Split — use the distributive property to break an awkward number into friendlier parts, multiply each part, then add (or subtract) the results.

A fast trick: multiplying by 99

To calculate 6 × 99 mentally, think of 99 as 100 − 1. Using the distributive property: 6 × 99 = 6 × 100 − 6 × 1 = 600 − 6 = 594. Numbers close to a multiple of 10 or 100 are much easier to multiply by after splitting them this way.

Worked Examples

  1. 1
    Use the commutative property to reorder the numbers so the friendly pair is together: 37 + 58 + 63 = 37 + 63 + 58.
  2. 2
    Use the associative property to add the friendly pair first: (37 + 63) + 58.
  3. 337 + 63 = 100.
  4. 4
    100 + 58 = 158.
Answer37 + 58 + 63 = 158
  1. 1Think of 99 as 100 − 1.
  2. 2
    Use the distributive property: 6 × 99 = 6 × (100 − 1) = (6 × 100) − (6 × 1).
  3. 36 × 100 = 600 and 6 × 1 = 6.
  4. 4
    600 − 6 = 594.
Answer6 × 99 = 594
  1. 1Left side: (12 × 5) × 2 = 60 × 2 = 120.
  2. 2Right side: 12 × (5 × 2) = 12 × 10 = 120.
  3. 3
    Both sides equal 120, even though the numbers were grouped differently.
  4. 4
    This shows the associative property of multiplication.
AnswerThis shows the associative property of multiplication — both sides equal 120

Diagram

The distributive property as an area model: splitting 7 × 12 into 7 × 10 and 7 × 2, then adding the two smaller products.

71027×12 = 70 + 14 = 84
⇔

3. Comparing and Ordering Whole Numbers

To compare whole numbers we look at the number of digits first — a number with more digits is always larger. If two numbers have the same number of digits we compare digit by digit starting from the left. We use the symbols > (greater than), < (less than) and = (equal to) to show the relationship between numbers. To order numbers we arrange them from smallest to biggest (ascending order) or biggest to smallest (descending order).

Colour key:> greater than< less than= equal tokey digit being compared

Rules for comparing whole numbers

1

Count the digits — a number with more digits is always larger. No digit-by-digit comparison is needed.

2

Same number of digits? Compare digit by digit from the left. The first position where the digits differ decides which number is larger.

3

All digits equal? Write = between the two numbers.

The comparison symbols

>

Greater than

The open end faces the larger number.

<

Less than

The pointed end faces the smaller number.

=

Equal to

Both numbers have exactly the same value.

Ascending and descending order

Ascending order — smallest to largest (numbers go up). Descending order — largest to smallest (numbers go down). Always start by counting digits to group numbers, then compare within each group digit by digit from the left.

Worked Examples

  1. 1
    Both numbers have 7 digits — so we compare digit by digit from the left.
  2. 2
    Millions place: both have 4 — equal.
  3. 3
    Hundred thousands place: both have 7 — equal.
  4. 4
    Ten thousands place: first number has 2, second has 3. Since 3 > 2, the second number is larger.
  5. 5
    Answer: 4 728 350 < 4 736 200 ✓
Answer4 728 350 < 4 736 200
  1. 1
    Count the digits in each number.
    987 654 has 6 digits — fewest digits, so it is the smallest.
    3 419 500 and 3 421 000 each have 7 digits.
    12 450 000 has 8 digits — most digits, so it is the largest.
  2. 2
    Compare the two 7-digit numbers: 3 419 500 vs 3 421 000.
    Millions: both have 3 — equal.
    Hundred thousands: both have 4 — equal.
    Ten thousands: 1 vs 2 — since 1 < 2, the number 3 419 500 is smaller.
  3. 3
    Ascending order: 987 654 < 3 419 500 < 3 421 000 < 12 450 000 ✓
Answer987 654 < 3 419 500 < 3 421 000 < 12 450 000

Diagram

Side-by-side digit comparison of 4 728 350 and 4 736 200 with each matching position aligned in columns and the differing ten thousands digits highlighted in orange with a less than symbol in red between the two numbers

MHThTThThHTU47283504736200<4 728 350 < 4 736 200
≈

4. Rounding Off Whole Numbers

Rounding off means replacing a number with a simpler number that is close to the original. We round to the nearest 10, 100, 1 000, 10 000, 100 000 or 1 000 000. To round off, look at the digit immediately to the right of the place you are rounding to. If that digit is 5 or more, round up. If that digit is 4 or less, round down — keep the digit the same and replace all digits to the right with zeros.

Colour key:rounding digitdigit to the rightround upround down

The two rounding rules

Round up — digit to the right is 5 or more

Increase the rounding digit by 1. Replace all digits to the right with zeros.

The digit to the right is 5, 6, 7, 8 or 9.

Round down — digit to the right is 4 or less

Keep the rounding digit the same. Replace all digits to the right with zeros.

The digit to the right is 0, 1, 2, 3 or 4.

How to round — step by step

1

Identify the rounding digit — find the digit in the place you are rounding to (e.g. the thousands digit if rounding to the nearest 1 000).

2

Look at the digit to the right — this is the digit immediately to the right of the rounding digit.

3

Apply the rule — if the digit to the right is 5 or more, round up. If it is 4 or less, round down.

4

Replace with zeros — write zeros in all places to the right of the rounding digit. Keep all digits to the left unchanged.

Key reminder

Only the digit immediately to the right of the rounding digit decides whether you round up or down. All other digits to the right simply become zeros.

Worked Examples

  1. 1
    We are rounding to the nearest million. The rounding digit is the millions digit — it is 3.
  2. 2
    Look at the digit immediately to the right — the hundred thousands digit is 8.
  3. 3
    Since 8 is 5 or more, we round up — increase the millions digit from 3 to 4.
  4. 4Replace all digits to the right with zeros.
  5. 5
    Answer: 3 847 215 rounded to the nearest million is 4 000 000 ✓
Answer3 847 215 rounded to the nearest million is 4 000 000
  1. 1
    We are rounding to the nearest ten thousand. The rounding digit is the ten thousands digit — it is 5.
  2. 2
    Look at the digit to the right — the thousands digit is 3.
  3. 3
    Since 3 is less than 5, we round down — keep the ten thousands digit as 5.
  4. 4Replace all digits to the right with zeros.
  5. 5
    Answer: 2 453 620 rounded to the nearest ten thousand is 2 450 000 ✓
Answer2 453 620 rounded to the nearest ten thousand is 2 450 000
  1. 1
    The rounding digit is the thousands digit — it is 6.
  2. 2
    Look at the hundreds digit — it is 5.
  3. 3
    Since 5 is 5 or more, we round up — increase the thousands digit from 6 to 7.
  4. 4Replace all digits to the right with zeros.
  5. 5
    Answer: 76 500 rounded to the nearest thousand is 77 000 ✓
Answer76 500 rounded to the nearest thousand is 77 000

Diagram

Number line showing 3 847 215 positioned between 3 000 000 and 4 000 000 with the midpoint at 3 500 000 marked and an arrow pointing to 4 000 000 to illustrate rounding up

3 000 0003 500 0004 000 0003 847 215rounds up3 847 215 rounds up to 4 000 000
×

5. Multiples and Factors

A multiple of a number is what we get when we multiply that number by 1, 2, 3, 4 and so on. The multiples of 3 are 3, 6, 9, 12, 15 and so on. A factor of a number is any number that divides into it exactly with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. The highest common factor (HCF) is the largest factor that two numbers share. The lowest common multiple (LCM) is the smallest multiple that two numbers share.

Colour key:multiplesfactorsHCFLCM

Key terms

Multiple

The result of multiplying a number by 1, 2, 3, 4 and so on.

Factor

Any number that divides into another number exactly with no remainder.

Highest Common Factor (HCF)

The largest factor that two or more numbers share.

Lowest Common Multiple (LCM)

The smallest multiple that two or more numbers share.

How to find all factors of a number

1

Start from 1 — Write 1 × the number. Both 1 and the number itself are always factors.

2

Work upwards — Try 2, 3, 4, … in order. If the number divides exactly, both the divisor and the quotient are factors.

3

Stop when factors meet — Stop when the two factors in a pair are equal or cross over. List all factors in ascending order.

HCF and LCM in everyday life

The HCF is useful when you want to split things into the largest equal groups possible. The LCM is useful when you want to find the first time two repeating events happen together at the same time.

Worked Examples

  1. 1Find all numbers that divide into 36 exactly.
  2. 2
    1 × 36 = 36 — so 1 and 36 are factors.
  3. 3
    2 × 18 = 36 — so 2 and 18 are factors.
  4. 4
    3 × 12 = 36 — so 3 and 12 are factors.
  5. 5
    4 × 9 = 36 — so 4 and 9 are factors.
  6. 6
    6 × 6 = 36 — so 6 is a factor (the pair meets here, so we stop).
  7. 7
    The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36 ✓
AnswerThe factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36
  1. 1
    List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  2. 2
    List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
  3. 3
    Find the common factors — numbers that appear in both lists: 1, 2, 3, 4, 6, 12.
  4. 4
    The largest of these common factors is 12.
  5. 5
    The HCF of 24 and 36 is 12 ✓
AnswerThe HCF of 24 and 36 is 12
  1. 1
    List the multiples of 4: 4, 8, 12, 16, 20, 24, …
  2. 2
    List the multiples of 6: 6, 12, 18, 24, …
  3. 3
    Find the smallest multiple that appears in both lists.
  4. 4
    The smallest multiple that appears in both lists is 12.
  5. 5
    The LCM of 4 and 6 is 12 ✓
AnswerThe LCM of 4 and 6 is 12

Diagram

Venn diagram comparing the factors of 24 and 36, with the common factors shown in the overlap and the highest common factor of 12 highlighted in orange

Factors of 24Factors of 36824918361234612HCF(24, 36) = 12
★

6. Prime and Composite Numbers

A prime number is a number that has exactly two factors — 1 and itself. Examples of prime numbers are 2, 3, 5, 7, 11 and 13. The number 1 is not prime because it only has one factor. A composite number is a number that has more than two factors. Examples are 4, 6, 8, 9 and 10. Every composite number can be written as a product of prime numbers — this is called prime factorisation.

Colour key:prime numberscomposite numbersprime factorsproduct notation

Key terms

Prime number

A number with exactly two factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13.

Composite number

A number with more than two factors. Examples: 4, 6, 8, 9, 10.

Prime factorisation

Writing a composite number as a product of its prime factors. Example: 36 = 2² × 3².

How to test whether a number is prime

1

Start from 2 — Try dividing the number by each prime: 2, 3, 5, 7, and so on.

2

Use divisibility shortcuts — even numbers are divisible by 2; if the digit sum is divisible by 3, so is the number; if it ends in 0 or 5, it is divisible by 5.

3

Stop when a factor is found — if any number divides in exactly, the number is composite. If no prime up to the square root divides in, the number is prime.

Remember: 1 is neither prime nor composite

The number 1 has only one factor (itself), so it does not meet the definition of a prime number or a composite number. The smallest prime number is 2 — the only even prime.

Worked Examples

  1. 1Check if any number other than 1 and 37 divides into 37 exactly.
  2. 2
    Try 2 — 37 is odd, so 2 does not divide into 37.
  3. 3
    Try 3 — digit sum: 3 + 7 = 10, which is not divisible by 3, so 3 does not divide into 37.
  4. 4
    Try 5 — 37 does not end in 0 or 5, so 5 does not divide into 37.
  5. 5
    Try 7 — 7 × 5 = 35, 7 × 6 = 42, so 7 does not divide exactly into 37.
  6. 6
    Since no number divides evenly into 37, it is a prime number ✓
Answer37 is a prime number
  1. 1
    Divide 36 by the smallest prime — 36 ÷ 2 = 18.
  2. 2
    Divide 18 by 2 — 18 ÷ 2 = 9.
  3. 3
    Divide 9 by the next prime — 9 ÷ 3 = 3.
  4. 4
    Divide 3 by 3 — 3 ÷ 3 = 1.
  5. 5
    The prime factors are 2, 2, 3, 3.
  6. 6
    Write the product notation: 36 = 2 × 2 × 3 × 3 = 2² × 3² ✓
Answer36 = 2 × 2 × 3 × 3 = 2² × 3²
  1. 1
    Check each number — 21 = 3 × 7, 22 = 2 × 11, both composite.
  2. 2
    23 has no factors other than 1 and 23 — prime ✓
  3. 3
    24, 25, 26, 27 and 28 are all composite.
  4. 4
    29 has no factors other than 1 and 29 — prime ✓
  5. 5
    30 is composite. 31 has no factors other than 1 and 31 — prime ✓
  6. 6
    32, 33, 34, 35 and 36 are all composite.
  7. 7
    37 has no factors other than 1 and 37 — prime ✓
  8. 8
    38 and 39 are composite.
  9. 9
    The prime numbers between 20 and 40 are 23, 29, 31 and 37 ✓
AnswerThe prime numbers between 20 and 40 are 23, 29, 31 and 37

Diagram

Factor tree for 36 showing prime factorisation — 36 splits into 2 and 18 then 18 into 2 and 9 then 9 into 3 and 3 with prime factors highlighted in orange and the final product notation 2 squared times 3 squared in blue

36218293336 = 2 × 2 × 3 × 3 = 2² × 3²
Next: Addition & Subtraction→
←Back to Grade 6