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The LCM From Prime Factors

A multiple contains the whole prime factorisation of the number. Build the LCM by taking each prime the most times it appears.

8 micro-lessons

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A Multiple Contains the Number

The prime factors of a multiple include all the prime factors of the number.

Look the other way (Textbook pp. 56–57)

Every factor is a subpart of a factorisation. Now compare a number with one of its multiples: 36 and $648 = 36 \times 18$ .

$36 = 2 \times 2 \times 3 \times 3$
$648 = (2 \times 2 \times 3 \times 3) \times (2 \times 3 \times 3)$

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Example 6: 14 and 35

Finding the smallest product that contains 2 × 7 and 5 × 7.

Problem. Find the LCM of 14 and 35. (Textbook p. 57, Example 6)

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Common Multiple or Not?

Decide whether a product contains both factorisations.

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Is $2 \times 3 \times 5 \times 7$ a common multiple of 14 and 35?

Build the LCM Prime by Prime

For each prime, take the largest number of times it appears.

The method (Textbook pp. 57–58)

List every prime that appears in any of the numbers.
For each prime, take the highest number of times it appears in any one factorisation.
Multiply.

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Example 7: 96 and 360

Counting 2s, 3s and 5s to build the LCM of 96 and 360.

Problem. Find the LCM of 96 and 360. (Textbook pp. 57–58, Example 7)

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HCF vs LCM Rule Card

Minimum for HCF, maximum for LCM.

HCF and LCM from prime factors

HCF: common primes, fewest times LCM: all primes, most times

What each part means

HCF

— only primes in every number, each the minimum number of times

LCM

— every prime that appears, each the maximum number of times

When it works

Both work for two or more numbers.

Example

96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3

360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5

HCF = 2 \times 2 \times 2 \times 3 = 24

LCM = 1440

LCM of 30 and 72

Fill the prime count table to find the LCM.

0 of 4 blanks filled

Figure it Out, Textbook p. 58 (a).

30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3.

Most 2s: . Most 3s: . Most 5s: .

LCM = 2 × 2 × 2 × 3 × 3 × 5 = .

← previousTorans, Gajak And The LCM next →Patterns And General Statements

A Multiple Contains the Number

The prime factors of a multiple include all the prime factors of the number.

Look the other way (Textbook pp. 56–57)

Every factor is a subpart of a factorisation. Now compare a number with one of its multiples: 36 and $648 = 36 \times 18$ .

$36 = 2 \times 2 \times 3 \times 3$
$648 = (2 \times 2 \times 3 \times 3) \times (2 \times 3 \times 3)$

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Common Multiple or Not?

Decide whether a product contains both factorisations.

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Is $2 \times 3 \times 5 \times 7$ a common multiple of 14 and 35?