Fundamental theorem of arithmetic
Every composite number is a product of primes in exactly one way, apart from order. Use the prime factors to find the HCF and LCM.
What the theorem says
A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11, ...). A composite number has more than two factors. The number 1 is neither prime nor composite. The theorem says every composite number breaks into primes in only one way: 360 = 2³ × 3² × 5, and no other set of primes multiplies to give 360.
How to write a prime factorisation
- 1Divide by the smallest prime that divides the number exactly: try 2, then 3, then 5, and so on.
- 2Keep dividing each quotient the same way until the quotient is 1.
- 3Write the number as the product of all the prime divisors you used.
- 4Group equal primes as powers, in ascending order, e.g. 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Example 1: HCF and LCM by prime factors
Find the HCF and LCM of 504 and 180 using the fundamental theorem of arithmetic. Check that HCF × LCM equals the product of the numbers.
- 504 = 2 × 2 × 2 × 3 × 3 × 7 = 2³ × 3² × 7
- 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
- HCF: common primes with their smallest powers = 2² × 3² = 36
- LCM: all primes with their greatest powers = 2³ × 3² × 5 × 7 = 2520
- Check: 36 × 2520 = 90720 and 504 × 180 = 90720.
HCF = 36 and LCM = 2520.
Example 2: can 12ⁿ end with 0?
Can the number 12ⁿ, where n is a natural number, end with the digit 0? Give a reason.
- A number ending in 0 is divisible by 10 = 2 × 5, so 5 must be one of its prime factors.
- 12ⁿ = (2² × 3)ⁿ = 2²ⁿ × 3ⁿ.
- The only primes in 12ⁿ are 2 and 3.
- By the fundamental theorem, this factorisation is unique, so 5 can never appear.
No. 12ⁿ never ends with the digit 0.
Two buses leave Koyambedu together, one every 12 minutes and one every 18 minutes. Since 12 = 2² × 3 and 18 = 2 × 3², they leave together again after LCM = 2² × 3² = 36 minutes.
2 marks: “Is 5 × 7 × 11 + 7 a composite number? Justify.” 5 × 7 × 11 + 7 = 7(55 + 1) = 7 × 56 = 392. It has factors other than 1 and itself, such as 7, so it is composite.
2 marks: “If 2ᵃ × 3ᵇ × 5ᶜ = 1350, find a, b and c.” 1350 = 2 × 675 = 2 × 3³ × 5², so a = 1, b = 3 and c = 2.
5 marks: “Find the HCF and LCM of 252, 378 and 504 by prime factorisation.” 252 = 2² × 3² × 7, 378 = 2 × 3³ × 7, 504 = 2³ × 3² × 7. HCF = 2 × 3² × 7 = 126. LCM = 2³ × 3³ × 7 = 1512. (For three numbers, HCF × LCM is not the product.)
HCF = low and common; LCM = high and all. HCF takes the lowest powers of common primes; LCM takes the highest powers of every prime.