Factors, multiples and primes
Use positive factors and positive multiples in HCF and LCM questions. Prime factorisation records all the prime factors, including repeats
- Find factors in pairs Collect both members of each exact product pair. Do not repeat a square-root factor \(\displaystyle \begin{aligned}22&=1\times22\\&=2\times11\\F_{22}&=\{1,2,11,22\}\end{aligned}\)
- Build positive multiples Multiply by successive positive whole numbers. The LCM is the first positive match between the lists \(\displaystyle \begin{aligned}9&:9,18,27,36,45\\15&:15,30,45\end{aligned}\)
- Classify by factors Among whole numbers greater than one, a prime has exactly two positive factors and a composite has more than two. Zero and one are neither. To test a whole number greater than one, try every prime no greater than its square root. If none divides exactly, the number is prime
- Use prime counts For the HCF, keep shared primes with their smaller counts. For the LCM, keep all required primes with their larger counts \(\displaystyle \begin{aligned}20&=2^2\times5\\70&=2\times5\times7\\\operatorname{HCF}(20,70)&=10\\\operatorname{LCM}(20,70)&=140\end{aligned}\)
Common mistakes
Treating one as prime
✕\(\displaystyle \begin{gathered}\text{Prime factors of }22:\\1,2,11\end{gathered}\)✓\(\displaystyle \begin{gathered}\text{Prime factors of }22:\\2,11\end{gathered}\)
One has only one positive factor. A prime needs exactly twoStopping too early
✕\(\displaystyle \begin{gathered}\text{Claimed final form:}\\52=2\times26\end{gathered}\)✓\(\displaystyle 52=2\times2\times13\)
The first product is equal to the number, but it still contains a composite factorChoosing a multiple for HCF
✕\(\displaystyle \operatorname{HCF}(14,35)=70\)✓\(\displaystyle \operatorname{HCF}(14,35)=7\)
A common factor divides each number. A common multiple is divisible by each numberDifficulty levels
introFind HCFs and LCMs of two small numbers by listing
standardCheck factors, positive multiples and prime status; find the HCF or LCM of two numbers
challengeClassify whole numbers, recognise complete prime factorisations, and find the HCF or LCM of two numbers
examChoose factor lists, prime classifications and prime factorisations using familiar number sizes
Questions about this topic
How do factors and multiples differ?+
A factor divides the number exactly. A multiple is made by multiplying the number by a whole number. HCF and LCM questions here use positive values
Why is one neither prime nor composite?+
It has only one positive factor. A prime has exactly two and a composite has more than two
Can I multiply the numbers to find their LCM?+
Their product is a common multiple. It is the lowest one only when the numbers have no shared prime factor
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 1: Number and financial mathematics
In this chapter
Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)