Prime Suspects

Prime Suspects

Every whole number bigger than 1 is built from prime numbers multiplied together, and there is only one way to build each one. Primes are the building blocks of the whole number system. Master how to break a number into its primes and you unlock a fast, reliable method for HCF and LCM: the kind examiners reward with full marks because you show working, not lucky guesses.

The words for it

Five terms, and the first one has two traps in it that examiners test every year:

Spot the prime

Exactly one of these is a prime number. Which one?

• 97
• 51
• 57
• 91

Factors and multiples

These two point in opposite directions, and mixing them up costs marks all the way to the LCM questions at the end:

Factor or multiple?

A number that divides exactly into 20 is a _____ of 20, for example 5. A number in the 20 times table, such as 60, is a _____ of 20. A number with exactly two factors, such as 5, is a _____ number, and writing a number as a product of those is called its prime _____.

Breaking it down

Prime factorisation means writing a number as a product of primes only, and a factor tree gets you there: keep splitting until every branch ends on a prime. Take 60. Split it any way you like: 60 = 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5. Every branch has reached a prime, so collect them: 2 × 2 × 3 × 5, which tidies to 60 = 2² × 3 × 5. Now start again from 60 = 4 × 15 instead. You get 2 × 2 × 3 × 5 again. That is not luck: it is a theorem, and it is the reason the HCF and LCM method at the end of this module works at all.

Order the factor-tree method

Put the steps of a prime factorisation into the right order, worked here on 84.

• Split 84 into any factor pair, for example 4 × 21
• Split each factor again: 4 = 2 × 2 and 21 = 3 × 7
• Stop each branch once it reaches a prime
• Write every prime as a product: 2 × 2 × 3 × 7
• Tidy repeated primes into index form: 2² × 3 × 7

Factorise it yourself

Find the largest prime factor of 273. (No calculator. Start by testing whether small primes divide it, and keep splitting.)

Match each number to its prime factorisation

HCF and LCM from primes

Once you have the prime factorisations, HCF and LCM fall out with no listing and no guessing. Take 24 = 2³ × 3 and 36 = 2² × 3². HCF (Highest Common Factor) takes the primes they share, each to the lowest power: 2² × 3 = 12. LCM (Lowest Common Multiple) takes every prime that appears, each to the highest power: 2³ × 3² = 72. Memory hook: HCF = common, lowest. LCM = all, highest. And a sanity check worth doing every time: the HCF cannot be bigger than the smaller number, and the LCM cannot be smaller than the bigger one.

Find the HCF

Find the HCF of 20 and 30. Factorise each one first, then take the primes they share, each to the lowest power.

Find the LCM

Same pair, 20 and 30. Find the LCM: every prime that appears, each to the highest power.

HCF or LCM?

Deciding which one a question wants is the whole skill. Four cases.

• A shop packs 48 red sweets and 36 blue sweets into identical bags, using every sweet with none left over. What is the greatest number of bags?
• Two buses leave the station together at 8am. One runs every 20 minutes, the other every 30. When do they next leave together?
• In general, what tells you which one a question is asking for?
• Back to the sweets. The shop makes its 12 bags. How many RED sweets are in each bag?

Which are true?

Select ALL THREE statements that are TRUE.

• Every whole number above 1 has exactly one prime factorisation, whichever factor pair you begin the tree with
• The HCF of two numbers is never larger than the smaller of them, and the LCM is never smaller than the larger
• If two numbers share no prime factors at all, their HCF is 1 and their LCM is simply their product
• 1 is prime, because it divides into every whole number
• A whole number is prime if it is odd
• The LCM of two numbers is always their product

Why 1 is not prime

In about 50 words, explain why mathematicians do not count 1 as a prime number. Include:

• how many factors a prime has, and how many 1 has
• what would go wrong with "every number has exactly one prime factorisation" if 1 were allowed to be prime
• which of those two reasons you think is the stronger, and why