IGCSE · Practice questions · Types of number: primes, factors, multiples, rational and irrational

Primes first, then everything.

Six original Cambridge-style questions on types of number: words and figures, primes, squares and cubes, prime factors, HCF and LCM, rational and irrational numbers and reciprocals.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Break it into primes, then build it back up.

Prime: exactly two factors. 1 is not prime; 2 is the only even prime.
Prime factors: divide by the smallest prime again and again; write the answer with indices.
HCF: primes in both numbers, at the lowest power.
LCM: every prime that appears, at the highest power.
Rational: can be written as \(\dfrac{a}{b}\); simplify roots before deciding.
Reciprocal: \(1 \div\) the number; make mixed numbers improper first.

Recall

[2 marks]

(a) Write 3 040 006 in words.

(b) Write twelve million five hundred thousand in figures.

(a) three million, forty thousand and six ✓
(b) 12 500 000 ✓

Recall

[2 marks]

\(1 \quad 2 \quad 9 \quad 15 \quad 23 \quad 27 \quad 49 \quad 64\)

From this list, write down (a) the prime numbers (b) the numbers that are both square numbers and cube numbers.

(a) 2 and 23 ✓
(b) \(1 = 1^{2} = 1^{3}\) and \(64 = 8^{2} = 4^{3}\), so 1 and 64. ✓

Calculation

[2 marks]

Write 252 as a product of its prime factors.

\(252 \div 2 = 126,\ 126 \div 2 = 63,\ 63 \div 3 = 21,\ 21 \div 3 = 7,\ 7 \div 7 = 1\) ✓
\(252 = 2^{2} \times 3^{2} \times 7\) ✓

Calculation

[3 marks]

\(120 = 2^{3} \times 3 \times 5\). Using your answer to the previous question, find (a) the HCF (b) the LCM of 252 and 120.

(a) Shared primes at the lowest power: \(2^{2} \times 3 = 12\) ✓
(b) All primes at the highest power: \(2^{3} \times 3^{2} \times 5 \times 7 = 2520\) ✓

Analysis

[3 marks]

(a) From \(\ \sqrt{0.25},\ \sqrt{18},\ 3.14,\ 2\pi\ \) write down the irrational numbers.

(b) Find the reciprocal of \(2\dfrac{3}{4}\).

(a) \(\sqrt{0.25} = 0.5 = \dfrac{1}{2}\) and \(3.14 = \dfrac{314}{100}\) are rational, so the irrational numbers are \(\sqrt{18}\) and \(2\pi\). ✓
(b) \(2\dfrac{3}{4} = \dfrac{11}{4}\), so the reciprocal is \(\dfrac{4}{11}\). ✓

Analysis

[3 marks]

Ama has 84 red beads and 120 blue beads. She shares all of them into identical bags, so every bag has the same number of red beads and the same number of blue beads, with none left over.

Find the largest number of bags she can make, and how many beads of each colour go in each bag.

The number of bags must divide both 84 and 120, and be as large as possible: the HCF. ✓
\(84 = 2^{2} \times 3 \times 7\), \(120 = 2^{3} \times 3 \times 5\), so HCF \(= 2^{2} \times 3 = 12\) bags. ✓
Each bag: \(84 \div 12 = 7\) red and \(120 \div 12 = 10\) blue beads. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.