6.1 Factors, Multiples & Prime Numbers

Factors, multiples and primes help explain how whole numbers are built. Use these ideas for equal grouping, common denominators and recurring cycles. Squares and cubes are taught in a separate section; order of operations there is an optional extension.

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Factors, Factor Pairs, and Highest Common Factor (HCF)

Finding Factor Pairs Systematically

A factor of an integer $n$ is a whole number that divides into $n$ leaving no remainder. Every factor has a partner factor, forming a factor pair ($a \times b = n$).

To find all factors of a number without omitting any, work systematically from $1$ upwards in pairs:

Finding all factors of 72:
 1 × 72 = 72
 2 × 36 = 72
 3 × 24 = 72
 4 × 18 = 72
 5 → Not a factor (72 does not end in 0 or 5)
 6 × 12 = 72
 7 → Not a factor (7 × 10 = 70, remainder 2)
 8 × 9  = 72  ← The factors meet! Stop here.

The Square Root Stopping Rule: You only ever need to test integers up to the square root of the number ($\sqrt{n}$). Since $\sqrt{72} \approx 8.48$, once you have tested $8$, all factor pairs have been discovered! The complete list of factors for $72$ is: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$ (12 factors in total).

Highest Common Factor (HCF / GCF)

The Highest Common Factor (HCF) of two or more numbers is the greatest whole number that divides into all of them without a remainder.

• Example: Find the HCF of 36 and 48:
  • Factors of $36$: $1, 2, 3, 4, 6, 9, \mathbf{12}, 18, 36$
  • Factors of $48$: $1, 2, 3, 4, 6, 8, \mathbf{12}, 16, 24, 48$
  • Common factors: $1, 2, 3, 4, 6, 12$
  • HCF = 12

Real-World Word Problem Archetype for HCF

Whenever a question asks to divide two or more collections of items into the largest possible number of identical groups with nothing left over, you are solving for HCF:

• Scenario: A caterer has $48$ mini sausage rolls and $60$ cheese straws. She wants to create identical platters with equal numbers of each item and no food remaining. What is the greatest number of platters she can prepare, and what will each platter contain?
  • $\text{HCF}(48, 60) = 12$ platters.
  • Sausage rolls per platter: $48 \div 12 = 4$.
  • Cheese straws per platter: $60 \div 12 = 5$.

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Multiples, Common Multiples, and Lowest Common Multiple (LCM)

Generating Multiples

A multiple of a number is the product of that number and a whole number. Zero is also a multiple; the lists here show positive multiples. Multiples form the times-tables of that number and extend to infinity:

• Multiples of $8$: $8, 16, 24, 32, 40, 48, 56, 64, 72, \dots$
• Multiples of $12$: $12, 24, 36, 48, 60, 72, 84, \dots$

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) is the smallest positive number that is a multiple of both (or all) numbers. In the lists above, the common multiples of $8$ and $12$ are $24, 48, 72, \dots$, so the LCM is 24.

Real-World Word Problem Archetype for LCM

Whenever a question describes two or more events operating on different cyclical schedules and asks when they will happen at the exact same time again, you are solving for LCM:

• Synchronised Flashing Lights: Lighthouse Beacon A flashes every $15$ seconds. Beacon B flashes every $20$ seconds. If both flash simultaneously at 20:00:00, at what time will they next flash together?
  • Multiples of $15$: $15, 30, 45, \mathbf{60}, 75, \dots$
  • Multiples of $20$: $20, 40, \mathbf{60}, 80, \dots$
  • $\text{LCM}(15, 20) = 60$ seconds ($1$ minute).
  • They will next flash together at 20:01:00.

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Prime Numbers and Prime Factor Decomposition

The Strict Definition of a Prime Number

A prime number is a whole number strictly greater than 1 that has exactly two distinct positive factors: $1$ and itself.

The Two Universal Traps

1. Why is the number 1 NOT a prime? The number $1$ has only one factor ($1$). By definition, a prime must have two distinct factors. Furthermore, if $1$ were prime, the Fundamental Theorem of Arithmetic (which states every number has a unique prime factorisation) would fail completely, because $12 = 2^2 \times 3 = 1 \times 2^2 \times 3 = 1^2 \times 2^2 \times 3$.
2. Why 2 is the ONLY even prime: Every even number greater than $2$ is divisible by $2$, giving it at least three factors ($1, 2,$ and itself). Therefore, $2$ is the smallest prime and the only even prime number.

The 25 Prime Numbers Below 100

Year 5 asks for recall of primes up to 19 and a method to establish whether a number up to 100 is prime. The list below is a checking reference:

 2,   3,   5,   7,  11,  13,  17,  19,  23,  29, 
31,  37,  41,  43,  47,  53,  59,  61,  67,  71, 
73,  79,  83,  89,  97

Beware of 'Imposter Primes'! These numbers look prime at a casual glance but are composite:

• $51$: $5 + 1 = 6$ (divisible by $3$; $3 \times 17 = 51$)
• $57$: $5 + 7 = 12$ (divisible by $3$; $3 \times 19 = 57$)
• $87$: $8 + 7 = 15$ (divisible by $3$; $3 \times 29 = 87$)
• $91$: Divisible by $7$ ($7 \times 13 = 91$)

Prime Factor Trees & Product of Prime Factors

Every composite number can be uniquely decomposed into a product of prime numbers. Use a factor tree to break a number down into prime branches:

          60
         /  \
        6    10
       / \   / \
     (2) (3)(2) (5)

 Prime factorisation of 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Finding HCF and LCM Using Prime Factor Venn Diagrams

To find the HCF and LCM of $60$ and $84$:

1. Decompose both into primes:
   • $60 = 2 \times 2 \times 3 \times 5$
   • $84 = 2 \times 2 \times 3 \times 7$
2. Draw a Venn diagram with two overlapping sets:
   • Shared prime factors, counting repeats: $2, 2, 3$
   • Factors left for 60: $5$
   • Factors left for 84: $7$
3. Calculate HCF: Multiply the numbers in the intersection: $$\text{HCF} = 2 \times 2 \times 3 = \mathbf{12}$$
4. Calculate LCM: Multiply ALL numbers in the Venn diagram: $$\text{LCM} = 5 \times (2 \times 2 \times 3) \times 7 = 5 \times 12 \times 7 = \mathbf{420}$$
5. Relationship check: $\text{HCF} \times \text{LCM} = a \times b \implies 12 \times 420 = 5,040 = 60 \times 84$.

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Establish whether a number below 100 is prime

Take 97. It is not even, its digits sum to 16 so it is not divisible by 3, it does not end in 0 or 5, and 7 × 13 = 91 while 7 × 14 = 98. Thus 2, 3, 5 and 7 do not divide it. A composite number below 100 has a prime factor below 10, because two factors both at least 10 would produce at least 100. The checks therefore establish that 97 is prime; recognising it from a memorised list is unnecessary.