Math · Unit 9

Integers and Number Theory

Negative numbers, factors, multiples, primes and roots

Extending numbers below zero seems like a small step and quietly breaks several rules that felt permanent - subtraction always making things smaller, for one.

The unit covers negative numbers and arithmetic with them, factors, multiples, primes, greatest common factor and lowest common multiple, and squares and roots.

This unit breaks down into 21 short steps and 120 questions, starting at difficulty 2 and building to 3. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.

Steps: 21
Questions: 120
Difficulty: 2-3

What this unit covers

Negative Numbers
GCF and LCM
Factors and Multiples
Prime and Composite Numbers
Squares and Square Roots

Where this fits

Follows Times Tables and Sharing. Pre-Algebra assumes all of it.

Where people slip

Two negatives multiplying to a positive is not arbitrary. It follows from keeping the distributive law working, and it is worth seeing why rather than memorising the rule.

How the unit is structured

Integers and Number Theory runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 2 and climb to 3 as you progress.

Step 1 · easierStep 21 · harder

Challenge rounds

Example questions

30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 120 in the unit altogether.

Factors and Multiples

Build the sentenceLevel 2
1. Build a true sentence about multiples.
Answer: 12 is a multiple of 4
4 x 3 = 12, so 12 is a multiple of 4.
Fact or fibLevel 2
2. 24 is a common multiple of 6 and 8.
Answer: True
6 x 4 = 24 and 8 x 3 = 24, so 24 is a multiple of both.
Fill the blankLevel 2
3. A number that divides another number exactly is called a ____.
- factorcorrect
- multiple
- product
- sum
A factor divides a number with no remainder.
Multiple choiceLevel 2
4. Which number is a factor of 12?
- 4correct
- 5
- 7
- 8
12 divided by 4 is exactly 3, so 4 is a factor of 12.
Choose all that applyLevel 3
5. Select all the numbers that are factors of 24.
- 6correct
- 8correct
- 5
- 9
6 and 8 both divide 24 evenly, while 5 and 9 do not.
Match the pairsLevel 3
6. Match each number to how many factors it has.
Answer: 9 = 3 factors; 12 = 6 factors; 7 = 2 factors
9 has 3 factors (1, 3, 9), 12 has 6 factors, and 7 has 2 factors.

GCF and LCM

Fact or fibLevel 2
7. The LCM of 4 and 6 is 12.
Answer: True
12 is the smallest number that both 4 and 6 divide into, so the LCM is 12.
Fill the blankLevel 2
8. The greatest number that divides both 12 and 18 exactly is ____.
- 6correct
- 3
- 2
- 9
6 goes into both 12 and 18, and nothing larger does, so the GCF is 6.
Build the sentenceLevel 3
9. Build a true sentence about the GCF.
Answer: The GCF of 8 and 12 is 4
4 divides both 8 and 12, and nothing larger does, so the GCF is 4.
Choose all that applyLevel 3
10. Select the common factors of 12 and 18.
- 2correct
- 3correct
- 4
- 5
12 and 18 share the factors 1, 2, 3, and 6, so 2 and 3 both qualify while 4 and 5 do not.
Guess the numberLevel 3
11. What is the LCM of 5 and 10?
Answer: 10
10 is already a multiple of 5, so the LCM of 5 and 10 is 10.
Match the pairsLevel 3
12. Match each pair of numbers to their GCF.
Answer: 16 and 24 = 8; 14 and 21 = 7; 20 and 30 = 10
GCF(16,24)=8, GCF(14,21)=7, and GCF(20,30)=10.

Negative Numbers

Fill the blankLevel 2
13. Numbers less than zero are called ____ numbers.
- negativecorrect
- positive
- prime
- whole
Numbers below zero, like -1 and -8, are negative numbers.
Put in orderLevel 2
14. Put these numbers in order from least to greatest.
Answer: -5 -> -2 -> 0 -> 3
From smallest to largest: -5, -2, 0, then 3.
Sort into groupsLevel 2
15. Sort each number as Positive or Negative.
Answer: -3 = Negative; 5 = Positive; -8 = Negative; 2 = Positive
Numbers below zero are negative and numbers above zero are positive.
Choose all that applyLevel 3
16. Select all numbers that are less than -3.
- -5correct
- -8correct
- -1
- 0
-5 and -8 are farther left than -3, while -1 and 0 are greater than -3.
Fact or fibLevel 3
17. The absolute value of -9 is 9.
Answer: True
Absolute value is the distance from zero, and -9 is 9 units from zero.
Multiple choiceLevel 3
18. What is -7 - 5?
- -12correct
- -2
- 12
- 2
Going 5 further below -7 lands on -12.

Prime and Composite Numbers

Build the sentenceLevel 2
19. Build a true sentence about prime numbers.
Answer: A prime number has exactly two factors
A prime number has exactly two factors: 1 and itself.
Fact or fibLevel 2
20. 9 is a prime number.
Answer: False
9 = 3 x 3, so it has more than two factors and is composite, not prime.
Fill the blankLevel 2
21. The number 1 is ____ prime nor composite.
- neithercorrect
- either
- both
- always
1 has only one factor, so it is neither prime nor composite.
Choose all that applyLevel 3
22. Select all the composite numbers.
- 15correct
- 21correct
- 17
- 23
15 = 3 x 5 and 21 = 3 x 7 are composite; 17 and 23 are prime.
Multiple choiceLevel 3
23. Which of these numbers is prime?
- 29correct
- 27
- 33
- 21
29 has only the factors 1 and 29, while 27, 33, and 21 are all composite.
Odd one outLevel 3
24. Three of these are composite numbers. Which one is the odd one out?
- 13correct
- 9
- 15
- 21
13 is prime, while 9, 15, and 21 are all composite.

Squares and Square Roots

Build the sentenceLevel 2
25. Build a true sentence about a square root.
Answer: The square root of 36 is 6
6 x 6 = 36, so the square root of 36 is 6.
Fact or fibLevel 2
26. The square of a negative number is negative.
Answer: False
A negative times a negative is positive, so (-3) x (-3) = 9, which is positive.
Fill the blankLevel 2
27. The result of a number multiplied by itself is called its ____.
- squarecorrect
- root
- factor
- double
5 x 5 = 25, so 25 is the square of 5.
Multiple choiceLevel 2
28. What is sqrt(49)?
- 7correct
- 6
- 8
- 24
7 x 7 = 49, so the square root of 49 is 7.
Odd one outLevel 2
29. Three of these are perfect squares. Which one is the odd one out?
- 20correct
- 16
- 25
- 36
16, 25, and 36 are perfect squares, but 20 is not the square of any whole number.
Match the pairsLevel 3
30. Match each square root to its value.
Answer: sqrt(16) = 4; sqrt(36) = 6; sqrt(81) = 9
sqrt(16)=4, sqrt(36)=6, and sqrt(81)=9.

Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.

How you practise

This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.

Build the sentence
Choose all that apply
Fact or fib
Fill the blank
Guess the number
Match the pairs
Multiple choice
Odd one out
Picture question
Put in order
Sort into groups
Spell it
True or false
Type the answer

Practise Integers and Number Theory

120 questions across 21 steps. Start with step one and crawl at your own pace.

Play this unit

Read about Integers and Number Theory

Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.

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