TVWO 1 · Chapter 2 · 2.2 HCF and LCM
Learn · Factors, primes, HCF and LCM
Learning goals: write a natural number greater than 1 as a product of prime factors, and calculate the HCF and LCM.
Theory A · Factors and prime numbers
In this chapter, the natural numbers are 0, 1, 2, 3, … . There is no largest natural number.
Factors
A positive factor divides a number exactly, without a remainder. For example, 3 is a factor of 24 because 24 ÷ 3 = 8.
Find factors in pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. For every positive natural number, 1 and the number itself are factors.
Prime numbers
A prime number has exactly two positive factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
1 is not prime: it has only one positive factor. 2 is the only even prime number.
A product of prime factors
Split a number into factors until every factor is prime. For 60:
60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5.
Check by multiplying your prime factors. In practice, type 2x2x3x5; keep repeated factors and do not use powers.
Theory B · HCF and LCM
Highest Common Factor (HCF)
The HCF is the greatest positive factor shared by the numbers.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
Common factors: 1, 2, 3, 6. So HCF(24, 30) = 6.
Quick method: start with the largest factor of the smallest number. Work down until you find one that divides every number exactly.
Example · HCF(12, 20)
Try factors of 12 from largest to smallest: 12, 6, 4, … .
12 and 6 do not divide 20 exactly. 4 does.
HCF(12, 20) = 4.
Lowest Common Multiple (LCM)
A multiple is made by multiplying a number by a natural number. The LCM is the smallest positive multiple shared by the numbers.
Positive multiples of 6: 6, 12, 18, 24, 30, … .
Positive multiples of 8: 8, 16, 24, 32, … .
So LCM(6, 8) = 24.
Quick method: start with the smallest positive multiple of the largest number. Work up until you find a multiple of every number.
Example · LCM(12, 20)
Try multiples of 20: 20, 40, 60, … .
20 and 40 are not multiples of 12. 60 is, because 60 ÷ 12 = 5.
LCM(12, 20) = 60.
Next step · Use prime factors
24 = 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
HCF: take the shared prime factors, using only the copies available in both numbers.
HCF(24, 60) = 2 × 2 × 3 = 12.
LCM: start with the prime factors of 24. Add only the factors from 60 that are still missing: one 5.
LCM(24, 60) = 2 × 2 × 2 × 3 × 5 = 120.
For three numbers, a factor must be shared by all three for the HCF. For the LCM, take enough copies of each prime to cover every number.
Which one do I need?
HCF: the largest equal groups or largest square tiles that fit exactly.
LCM: the first time repeating events happen together again.
Two lights flash every 30 and 40 seconds. When do they next flash together?
Use the LCM: multiples of 40 are 40, 80, 120. The first divisible by 30 is 120. They next flash together after 120 seconds.