Types of Number
Tested on Papers 1 & 3
Numbers come in families, and the exam expects you to know each one by name:
Natural numbers: 1, 2, 3, … (counting numbers). Integers: …, −2, −1, 0, 1, 2, … (whole numbers including negatives). Rational numbers: any number that can be written as a fraction of integers — includes all fractions, all terminating decimals, and recurring decimals. Irrational numbers: cannot be written as a fraction — like \(\sqrt{5}\) and \(\pi\); their decimals run forever without repeating.
Square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…) and cube numbers (1, 8, 27, 64, 125…) should be instant recall. The reciprocal of a number is 1 divided by it — flip the fraction: the reciprocal of \(\tfrac{2}{3}\) is \(\tfrac{3}{2}\).
HCF (highest common factor): the biggest number dividing both. LCM (lowest common multiple): the smallest number both divide into. Small numbers → list factors/multiples. Bigger numbers → use the prime factorisations.
Write 60 as a product of its prime factors, in index form.
Use the factor tree above: 60 = 6 × 10 = (2 × 3) × (2 × 5). Collect the primes: 2, 2, 3, 5.
\( 60 = 2^{2} \times 3 \times 5 \)
“Index form” means using powers: write 2² rather than 2 × 2.
Find the HCF and LCM of 24 and 36.
Prime DNA: \(24 = 2^{3} \times 3\) and \(36 = 2^{2} \times 3^{2}\).
HCF — take the lower power of each shared prime: \(2^{2} \times 3 = 12\).
LCM — take the higher power of every prime that appears: \(2^{3} \times 3^{2} = 8 \times 9 = 72\).
Check: HCF × LCM = 12 × 72 = 864 = 24 × 36 ✔ (always true for two numbers).
From the list −5, \(\tfrac{7}{2}\), \(\sqrt{5}\), 0.36, \(\pi\) write down (a) the integers, (b) the irrational numbers.
(a) Integers: only −5. (b) Irrational: \(\sqrt{5}\) and \(\pi\) — everything else can be written as a fraction (0.36 = \(\dfrac{36}{100}\)).
Write down the reciprocal of (a) \(\tfrac{2}{3}\), (b) 0.2.
(a) Flip it: \(\tfrac{3}{2}\). (b) 0.2 = \(\tfrac{1}{5}\), so its reciprocal is 5.
Check: a number × its reciprocal = 1. 0.2 × 5 = 1 ✔
Q1. Write 84 as a product of its prime factors, in index form.
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84 = 22 × 3 × 7 ✓ A1
Q2. Find the HCF and the LCM of 18 and 30.
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HCF = 2 × 3 = 6 ✓ A1
LCM = 2 × 32 × 5 = 90 ✓ A1
Q3. Which of these is irrational? \(\sqrt{9}\) , \(\sqrt{7}\) , \(\tfrac{22}{7}\) , 0.5
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√7 is irrational ✓ B1
Q4. Write down the reciprocal of \(1\tfrac{1}{4}\).
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Q5. From the list 8, 16, 27, 36, 49, 64 write down (a) all the square numbers, (b) the number that is both a square number and a cube number.
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(b) 64 = 82 = 43 ✓ B1
Q6. Two Gaborone combi routes leave the rank together at 07:00. Route A leaves every 12 minutes, Route B every 15 minutes. At what time do they next leave together?
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LCM = 22 × 3 × 5 = 60 minutes ✓ A1
07:00 + 60 min = 08:00 ✓ A1
Q7. Write down a prime number between 90 and 100.
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- Forgetting index form: writing 84 = 2 × 2 × 3 × 7 when the question says “index form” drops a mark — write 2² × 3 × 7.
- HCF/LCM swapped: if your “HCF” is bigger than both numbers, it’s the LCM. The HCF can never exceed the smaller number.
- √ of a square number called irrational: \(\sqrt{16}=4\), \(\sqrt{81}=9\) — rational. Only roots of non-squares are irrational.
- Reciprocal of a mixed number without converting: always make it an improper fraction first, then flip.
C1.1 appears early in Papers 1 and 3 as quick marks: “write down a prime”, “express as a product of primes”, “find the HCF/LCM”. The word problems (combis, flashing lights, packs of items) hide LCM/HCF inside a story — spot the pattern and the marks are easy. Extended papers assume all of this without warning.
(then move on to N-C2 Sets and Venn Diagrams)