MathematicsExam code: 0580

Prime Factors, HCF and LCM

Number

PractiseFlashcards

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

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