IGCSE Mathematics: Types of Number Practice Questions
Every integer greater than 1 can be written as a product of prime factors in exactly one way. That prime factorisation is what makes finding the highest common factor and the lowest common multiple quick and reliable.
Topic 1.1 of Cambridge IGCSE Mathematics 0580 opens Paper 2 more often than any other topic. HCF and LCM questions carry method marks for the prime factorisation itself, so writing it out is worth doing even when you can spot the answer. The questions below use that method throughout.
What you need to know for Types of Number
- Prime numberA number with exactly two factors, 1 and itself. 1 is not prime because it has only one factor. 2 is the only even prime.
- Prime factorisationWriting a number as a product of primes, usually with index notation. Every integer above 1 has exactly one prime factorisation.
- Highest common factorTake the prime factors common to both numbers, each raised to the lower of the two powers, and multiply.
- Lowest common multipleTake every prime factor appearing in either number, each raised to the higher of the two powers, and multiply.
- Squares, cubes and rootsSquare numbers are 1, 4, 9, 16, 25 and so on. Cube numbers are 1, 8, 27, 64, 125. Learn squares to 15 and cubes to 5.
- Rational and irrationalA rational number can be written as a fraction of two integers, including all terminating and recurring decimals. An irrational number cannot, such as pi and the square root of 2.
IGCSE Mathematics Types of Number questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Write 360 as a product of its prime factors, using index notation.
Show the worked answer
- Divide repeatedly by the smallest prime that goes in: 360 divided by 2 = 180, then 90, then 45.
- 45 is not divisible by 2, so move to 3: 45 divided by 3 = 15, then 15 divided by 3 = 5.
- 5 is prime, so stop. The prime factors are 2, 2, 2, 3, 3 and 5.
- In index notation: 360 = 23 x 32 x 5. Check by multiplying: 8 x 9 x 5 = 360.
Find the highest common factor and the lowest common multiple of 24 and 60.
Show the worked answer
- Write each as a product of primes: 24 = 23 x 3 and 60 = 22 x 3 x 5.
- For the HCF, take each common prime to the lower power: 22 x 3 = 4 x 3 = 12.
- For the LCM, take each prime that appears in either, to the higher power: 23 x 3 x 5 = 8 x 3 x 5 = 120.
- Check: 24 x 60 = 1440 and HCF x LCM = 12 x 120 = 1440, which confirms both answers.
Two lighthouses flash at regular intervals. One flashes every 12 seconds and the other every 18 seconds. They flash together at 8:00 pm exactly. Find the next time they flash together.
Show the worked answer
- They flash together at times that are multiples of both 12 and 18, so the interval between coincidences is the LCM of 12 and 18.
- 12 = 22 x 3 and 18 = 2 x 32.
- LCM = 22 x 32 = 4 x 9 = 36 seconds.
- They therefore flash together again 36 seconds after 8:00 pm, at 8:00:36 pm.
From the list 4, 7, 9, 11, 16, 27, 30, write down all the prime numbers, all the square numbers and all the cube numbers.
Show the worked answer
- Primes have exactly two factors. 7 and 11 qualify. 4, 9, 16, 27 and 30 all have more than two factors.
- Square numbers are the result of multiplying an integer by itself: 4 = 22, 9 = 32, 16 = 42.
- Cube numbers are the result of an integer cubed: 27 = 33. Note that 1 would also count as both a square and a cube if it appeared in the list.
- 30 belongs to none of the three categories.
Common mistakes in this topic
- Treating 1 as a prime number.
- Confusing HCF with LCM. The HCF is never larger than the smaller number, and the LCM is never smaller than the larger one.
- Forgetting index notation when the question asks for it.
- Assuming all odd numbers are prime.
- Missing that 2 is a prime number.
Exam tips
- Always write both prime factorisations before finding an HCF or LCM. That line is worth a method mark on its own.
- Check your answers using HCF multiplied by LCM equals the product of the two numbers.
- For word problems, decide first whether events are coming together, which needs the LCM, or being divided into equal groups, which needs the HCF.
- Learn square numbers up to 15 squared and cube numbers up to 5 cubed by heart.
- A number is rational if you can write it as a fraction of integers. Recurring decimals are rational, so 0.333 recurring is not irrational.
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Types of Number FAQs
How do I find the HCF and LCM using prime factors?
Write both numbers as products of primes in index form. For the highest common factor, multiply the primes common to both, each taken to the lower power. For the lowest common multiple, multiply every prime that appears in either number, each taken to the higher power.
Is 1 a prime number?
No. A prime number has exactly two distinct factors, itself and 1. The number 1 has only one factor, so it is not prime. It is also not composite. The smallest prime number is 2, which is also the only even prime.
When do I use HCF and when do I use LCM?
Use the LCM when repeating events come back together, such as two buses departing at the same moment. Use the HCF when something is being split into the largest possible equal groups, such as the biggest identical bundles you can make from two different quantities.
What is the difference between rational and irrational numbers?
A rational number can be written as a fraction of two integers, which includes all integers, terminating decimals and recurring decimals. An irrational number cannot be written that way, and its decimal expansion never terminates or repeats. Pi and the square root of 2 are irrational.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.