Natural numbers, integers, primes, squares, cubes, factors, multiples, rational and irrational numbers and reciprocals: the vocabulary every other topic in IGCSE Mathematics is written in.
Every whole number greater than 1 can be written as a product of primes in exactly one way. Write both numbers as prime products: the HCF takes the primes they share (lowest powers), the LCM takes every prime that appears (highest powers).
Fig. 1: a factor tree for 72. Split until every branch ends in a prime (circled), then collect them: 72 = 2 × 2 × 2 × 3 × 3
Section 01
Naming the numbers.
In the lab, pick two numbers and watch their multiples light up on the 1 to 100 grid; then switch to factors and step through the sieve until only the primes are left. Notice which numbers are never lit: they are the primes.
Natural numbers are the counting numbers \(1, 2, 3, \dots\). Integers are the whole numbers: positive, zero and negative, such as \(-4, 0, 17\).
A prime has exactly two factors, 1 and itself: \(2, 3, 5, 7, 11, 13, \dots\). 1 is NOT prime (it has only one factor).
Square numbers are \(1, 4, 9, 16, 25, \dots\) (\(n \times n\)); cube numbers are \(1, 8, 27, 64, 125, \dots\) (\(n \times n \times n\)).
A factor of 12 divides into it exactly (1, 2, 3, 4, 6, 12); a multiple of 12 is in its times table (12, 24, 36, ...).
Worked example Sorting a list
From the list \(\ 1,\ 2,\ 9,\ 15,\ 27,\ 31,\ 64\ \) write down (a) the prime numbers (b) the square numbers (c) the cube numbers.
(a) Test each number for factors other than 1 and itself: 9, 15, 27 and 64 all have extra factors; 1 has only one factor so it is not prime.
(a) Write six billion in figures. (b) Write 10 007 in words. (c) Write 2 040 500 in words.
(a) One billion is a thousand million, \(1\,000\,000\,000\) (9 zeros).
(b) Split into thousands and the rest: 10 thousand and 7.
(c) Split into groups of three from the right: 2 | 040 | 500, so 2 million, 40 thousand, 500.
Answer: (a) 6 000 000 000 (b) ten thousand and seven (c) two million, forty thousand, five hundred
Examiner trap
1 is not a prime number, and 2 is the only even prime. A list of primes starting \(1, 3, 5, 7, 9\) loses marks twice: 1 and 9 are not prime, and 2 is missing.
Section 02
Prime factors.
Open the prime factors tab, choose a number and press Next step: the lab divides by the smallest prime that works, again and again, until 1 is left. The primes it divided by are the prime factors.
Worked example Product of prime factors
Write 360 as a product of its prime factors, using index notation.
Divide by the smallest prime that goes in: \(360 \div 2 = 180\), \(180 \div 2 = 90\), \(90 \div 2 = 45\).
2 no longer goes into 45, so try 3: \(45 \div 3 = 15\), \(15 \div 3 = 5\).
5 is prime: \(5 \div 5 = 1\). Stop.
Collect the divisors: three 2s, two 3s and one 5.
Answer: \(360 = 2^{3} \times 3^{2} \times 5\)
Examiner trap
\(72 = 8 \times 9\) is true, but it is not a product of PRIMES: 8 and 9 both split further. Keep going until every factor is prime, then use index notation: \(2^{3} \times 3^{2}\).
Stage 1 · Learn
Check your understanding
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Which of these numbers is prime?
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Which of these numbers is irrational?
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What is the reciprocal of 0.2?
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Which is 72 written as a product of its prime factors?
Section 03
HCF and LCM.
The highest common factor (HCF) is the biggest number that divides into both; the lowest common multiple (LCM) is the smallest number both divide into. Prime factors give both at once.
Do not mix the rules. The HCF can never be bigger than the smaller number, and the LCM can never be smaller than the bigger one. If your HCF of 84 and 90 is 1260, you have swapped them.
Section 04
Rational, irrational and reciprocals.
A rational number can be written as a fraction \(\dfrac{a}{b}\) of two integers (\(b \ne 0\)). Terminating decimals, recurring decimals and square roots of square numbers are all rational. An irrational number cannot be written as a fraction: its decimal goes on for ever with no repeating pattern.
Irrational: \(\pi\), \(\sqrt{2}\), \(\sqrt{7}\), \(2 + \sqrt{3}\): the square root of any whole number that is not a square number is irrational.
The reciprocal of a number is 1 divided by it: the reciprocal of 5 is \(\dfrac{1}{5}\), and the reciprocal of \(\dfrac{a}{b}\) is \(\dfrac{b}{a}\). A number times its reciprocal is 1. Zero has no reciprocal.
Worked example Reciprocals
Find the reciprocal of (a) 2.5 (b) \(\dfrac{3}{4}\) (c) \(1\dfrac{2}{3}\).
(a) \(2.5 = \dfrac{5}{2}\), so the reciprocal is \(\dfrac{2}{5} = 0.4\).
(b) Turn the fraction upside down: \(\dfrac{4}{3}\).
(c) First make it improper: \(1\dfrac{2}{3} = \dfrac{5}{3}\), then turn it over: \(\dfrac{3}{5}\).
\(\sqrt{9}\) looks irrational because it has a root sign, but \(\sqrt{9} = 3\), which is rational. Always simplify first. And the reciprocal of \(1\dfrac{2}{3}\) is not \(1\dfrac{3}{2}\): convert to an improper fraction before turning it over.
Stage 2 · Exam
Exam-style questions
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Find the highest common factor (HCF) of 60 and 84.
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Two lights flash together at exactly 9 pm. One flashes every 15 seconds, the other every 20 seconds. After how many seconds do they next flash together?
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\(P = 2^{3} \times 3 \times 5^{2}\) and \(Q = 2^{2} \times 3^{2} \times 5\). Find the lowest common multiple of \(P\) and \(Q\).
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Which list contains only irrational numbers?
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