Prime Factorisation, HCF and LCM

A prime number has exactly two factors: 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. 1 is not a prime number, because it has only one factor.

Every whole number bigger than 1 can be made by multiplying primes together. For example, 42 = 2 \times 3 \times 7. This is called writing 42 as a product of its prime factors. Product means the numbers are multiplied.

In the exam you are usually asked to write a number as a product of its prime factors, for 2 marks. The next part often asks for the highest common factor (HCF) or the lowest common multiple (LCM) of two numbers, for 2 more marks. Some questions give two numbers already written with powers, such as 2^3 \times 5, and ask for their HCF or LCM.

Listing factors and multiples works well for small numbers. Prime factors are quicker and safer for bigger numbers, and this page shows how.

A factor tree splits a number into two factors, then splits those factors, until every branch ends in a prime. Circle each prime as you reach it, so you know that branch has stopped.

You can start with any factor pair except 1 and the number itself. Starting with 140 = 2 \times 70 or 140 = 4 \times 35 gives the same primes in the end: 2, 2, 5 and 7. Every number has only one set of prime factors.

Write the answer as a product, with multiplication signs, smallest prime first: 140 = 2 \times 2 \times 5 \times 7. A list such as 2, 2, 5, 7 is not a product. Check by multiplying back: 2 \times 2 \times 5 \times 7 = 140

Another way is to keep dividing by a prime that goes in exactly: 140 \div 2 = 70, then 70 \div 2 = 35, then 35 \div 5 = 7. The primes you divided by, and the 7 left at the end, are the prime factors.

When a prime appears more than once, write it as a power. 2 \times 2 \times 2 is 2^3, said 2 cubed or 2 to the power 3. The small number, the index, counts how many times the prime is multiplied.

So 140 = 2 \times 2 \times 5 \times 7 = 2^2 \times 5 \times 7. A prime that appears only once has no power written: 5 means 5^1

Edexcel asks for this as a product of powers of its prime factors. Write each prime once, smallest first, with its power.

Some questions give the form, such as 2^a \times 3^b, and ask for the values of a and b. Count how many 2s and how many 3s there are.

Write both numbers as products of prime factors first. For example, 42 = 2 \times 3 \times 7 and 70 = 2 \times 5 \times 7

Draw two overlapping circles, one for each number. A prime that is in both numbers goes in the overlap, once. The primes left over go in their own circle, outside the overlap.

The HCF is the product of the primes in the overlap: 2 \times 7 = 14. The LCM is the product of every prime in the diagram: 3 \times 2 \times 7 \times 5 = 210

When a prime is repeated, count carefully. Put it in the overlap as many times as it appears in both numbers, and put any extra copies in their own circle.

Check your answers. The HCF divides into both numbers: 42 \div 14 = 3 and 70 \div 14 = 5. The LCM is in both times tables: 210 \div 42 = 5 and 210 \div 70 = 3

Some questions give the numbers already as products of powers, for example A = 2^3 \times 7 and B = 2^2 \times 3 \times 7. You don't need to work out A and B first.

• HCF: take each prime that is in both numbers, with the smaller power.
• LCM: take every prime that is in either number, with the bigger power.

Here 2 and 7 are in both. The smaller powers give HCF = 2^2 \times 7 = 28. Every prime with its bigger power gives LCM = 2^3 \times 3 \times 7 = 168

In fact A = 56 and B = 84, the numbers from worked example 4, so the LCM is 168 again. This is the Venn diagram method written with powers.

Some questions don't say HCF or LCM. You have to decide which one you need.

• HCF: splitting things into the largest possible equal groups or pieces, with nothing left over. For example, the longest equal lengths cut from two ropes, or the biggest identical bags made from two sets of sweets.
• LCM: things that repeat, to find when they happen together again. For example, two lights flashing, or packs of different sizes when you want the same number of each.

A quick check: the HCF is never bigger than the smaller number, and the LCM is never smaller than the bigger number.

• Stopping before every branch ends in a prime. Numbers such as 21, 39 and 51 are not prime, so keep splitting until every branch ends in a prime.
• Putting 1 in a factor tree or an answer. 1 is not prime, so never split a number as 1 times itself.
• Writing a list instead of a product. Join the primes with multiplication signs, never commas or plus signs.
• Leaving out the powers. A product of powers means writing 2^3, not 2 \times 2 \times 2
• Mixing up the HCF and the LCM. The HCF is never bigger than the smaller number; the LCM is never smaller than the bigger number.
• Giving one prime as the HCF or LCM. Multiply every prime in the overlap for the HCF, and every prime in the diagram for the LCM.