Prime Factors, HCF and LCM

Mathematics revision notes · Cambridge (CIE) · 0580

BackFlashcards

1. Primes and prime factors

Prime numbers

  • Prime numbers are those divisible by nothing except themselves and 1, giving them a factor count of exactly two
  • The primes begin 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and continue without end
  • Every even number beyond 2 has 2 as a third factor, which leaves 2 as the sole even prime
  • 1 is not prime, because it has a single factor rather than two

Prime factors

  • Broken down far enough, any integer greater than 1 is a product of primes alone, and those primes are its prime factors
    • 30 is built from 2, 3 and 5, since 2 × 3 × 5 = 30
  • Every integer above 1 has exactly one set of prime factors, whatever route is taken to find them
  • This uniqueness is what allows prime factors to be used for comparing two numbers

2. Writing a number as a product of prime factors

Building a factor tree

  • Split the number into any factor pair, then split each factor again
  • Continue down every branch until only primes remain, circling each prime as it appears
  • Dividing repeatedly by the smallest prime that fits produces a prime at every stage and keeps the tree tidy
A factor tree for 432 built by dividing by the smallest prime each time: 432 splits into a circled 2 and 216, 216 into a circled 2 and 108, 108 into a circled 2 and 54, 54 into a circled 2 and 27, 27 into a circled 3 and 9, and 9 into two circled 3s, so every step drops one prime out to the left
Source: Prime factor decomposition by Save My Exams
  • Splitting by a convenient number such as 10 is equally valid, but non-prime results must then be split further
The same number 360 split a different way: 360 branches into 10 and 36, 10 into a circled 2 and a circled 5, 36 into 6 and 6, and each 6 into a circled 2 and a circled 3, ending on the same collection of primes as any other route
Source: Prime factor decomposition by Save My Exams
  • The circled primes are the same whichever pairs were chosen
Factor tree for 360: 360 branches into a circled 2 and 180, 180 branches into a circled 2 and 90, 90 into a circled 2 and 45, 45 into a circled 3 and 15, and 15 into a circled 3 and a circled 5
Source: Prime factor decomposition by Save My Exams

Writing the answer

  • Multiply the circled primes together, writing them in ascending order
    • 252 = 2 × 2 × 3 × 3 × 7
  • Collect repeated primes into index form, which is the form to give whenever a question asks for index form or the simplest form
    • 252 = 2² × 3² × 7

Worked example

A number as a product of its prime factors

Write 360 as a product of its prime factors, in index form.

Solution:

  • Split 360 into any two factors and keep splitting until every branch ends on a prime
  • 360 = 36 × 10
  • 36 = 6 × 6, and each 6 = 2 × 3
  • 10 = 2 × 5
  • The primes at the ends of the branches are 2, 3, 2, 3, 2, 5
  • Collect them in ascending order: 2 × 2 × 2 × 3 × 3 × 5
  • In index form, 360 = 2³ × 3² × 5
  • Check by multiplying back: 8 × 9 × 5 = 360
  • Starting from a different split, such as 360 = 4 × 90, gives the same answer: the tree's shape does not matter

Worked example

A product of primes from a number already in primes

N = 2⁴ × 3 × 7. Write 18N² as a product of its prime factors, in index form.

Solution:

  • There is no need to work out N: use its indices
  • Squaring doubles every index (3 and 7 count as 3¹ and 7¹): N² = 2⁸ × 3² × 7²
  • Only N is squared, not the 18, so write 18 in primes on its own: 18 = 2 × 3²
  • Multiplying adds the indices of each prime: 2⁸ × 2 = 2⁹, 3² × 3² = 3⁴, and 7² is unchanged
  • 18N² = 2⁹ × 3⁴ × 7²

3. Highest common factor and lowest common multiple

What each one means

  • Where a single value divides two numbers exactly, it is a common factor of the pair, and 1 always qualifies
  • The highest common factor, or HCF, is the largest such value
  • A value sitting in both numbers' times tables is a common multiple
  • Of the infinitely many such values, the smallest is the lowest common multiple, or LCM
  • The HCF is what cancels a fraction to its simplest form; the LCM is what supplies a common denominator

Finding them by listing

  • List every factor of both numbers, then take the largest value common to the two lists
  • List the multiples of both, then take the smallest value common to the two lists
  • Listing is quick for small numbers and unreliable for large ones

Finding them from prime factors

  • Decompose both numbers into primes first
  • Place the primes the two numbers share in the overlap of a Venn diagram, taking each shared prime only as many times as it appears in both
  • Place the remaining primes in the outer region belonging to their own number
  • Multiplying the overlap gives the HCF; multiplying everything in the whole diagram gives the LCM
Venn diagram for the prime factors of 42 and 90: 7 sits in the region for 42 alone, 3 and 5 sit in the region for 90 alone, and 2 and 3 sit in the overlap shared by both
Source: Highest common factor (HCF) by Save My Exams
  • Where no primes are shared, the overlap holds a 1 and the HCF is 1

Worked example

Finding the HCF and LCM from prime factors

Find the highest common factor and the lowest common multiple of 60 and 144.

Solution:

  • Decompose each into primes
  • 60 = 2² × 3 × 5
  • 144 = 2⁴ × 3²
  • Both contain 2 and 3. Take the lower index of each for the shared part: 2² and 3
  • HCF = 2² × 3 = 12
  • For the LCM collect every prime occurring anywhere, each at its highest index: 2⁴, 3² and 5
  • LCM = 16 × 9 × 5 = 720

Exam tip

Keep the prime factor working on the page

Before you give an HCF or LCM, write both numbers as products of primes, or draw a factor tree or division table for each, splitting right down to primes. That working earned a mark on its own even when the final answer was wrong. The same holds when a single number has to be written as a product of primes: a correct tree or table earned a mark there too.

Worked example

HCF and LCM of algebraic terms

Find the highest common factor and the lowest common multiple of 20p⁴q² and 30p³q⁵.

Solution:

  • Deal with the number part and each letter separately
  • 20 = 2² × 5 and 30 = 2 × 3 × 5, so the numbers have HCF 2 × 5 = 10 and LCM 2² × 3 × 5 = 60
  • For the HCF, each letter takes its lower power: p³ and q²
  • HCF = 10p³q²
  • For the LCM, each letter takes its higher power: p⁴ and q⁵
  • LCM = 60p⁴q⁵
  • Check: 20p⁴q² = 10p³q² × 2p and 30p³q⁵ = 10p³q² × 3q³, and 2p and 3q³ share no factor, so no larger term divides both

4. Using prime factor decomposition

Recognising squares and cubes

  • A number is a square number exactly when every index in its prime factor decomposition is even
    • 1225 = 5² × 7², so it is a square
  • A number is a cube number exactly when every index is a multiple of 3
    • 3375 = 3³ × 5³, so it is a cube
  • This test settles the question without any square rooting

Finding a square root from prime factors

  • Halve every index, then multiply the result out
    • 1225 = 5² × 7², so its square root is 5 × 7 = 35
  • The same idea works for a cube root by dividing every index by 3
    • 3375 = 3³ × 5³, so its cube root is 3 × 5 = 15

Finding an exact square root that is not an integer

  • Where the indices are not all even, split off the largest even part of each
  • Group the even-indexed primes together, since those come out of the root whole
  • Whatever is left stays under the root sign as a surd
    • 300 = 2² × 3 × 5², which regroups as (2² × 5²) × 3
    • The square part gives 2 × 5 = 10, leaving √300 = 10√3
  • This is how an exact answer is produced on the non-calculator papers, where a decimal would not be accepted as exact

Worked example

A square root from prime factors

Given that 3600 = 2⁴ × 3² × 5², find √3600 without a calculator.

Solution:

  • A square root halves every index, so pair the primes up
  • 2⁴ becomes 2², 3² becomes 3¹, and 5² becomes 5¹
  • √3600 = 2² × 3 × 5
  • 4 × 3 × 5 = 60
  • Every index in 3600 is even, which is exactly what makes it a square number
  • Check: 60 × 60 = 3600