The Fundamental Theorem of Arithmetic
Every composite number can be broken into a product of primes:
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
1001 = 7 × 11 × 13
The Fundamental Theorem of Arithmetic says:
Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
So, however you start factorising 60, you always end with two 2s, one 3 and one 5.
Example. Can 6ⁿ ever end in the digit 0 for a natural number n?
A number ends in 0 only if it is divisible by 10 = 2 × 5, so its prime factorisation must contain 5. But 6ⁿ = 2ⁿ × 3ⁿ, whose only primes are 2 and 3. By uniqueness, 5 can never appear. So 6ⁿ never ends in 0.
HCF and LCM by prime factorisation
Write each number as a product of prime powers. Then:
HCF = product of the smallest power of each common prime.
LCM = product of the greatest power of every prime involved.
Example. Find the HCF and LCM of 96 and 404.
96 = 2⁵ × 3 and 404 = 2² × 101.
HCF = 2² = 4
LCM = 2⁵ × 3 × 101 = 9696
Example with three numbers. 6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5.
HCF = 2¹ × 3¹ = 6
LCM = 2³ × 3² × 5 = 360
The HCF × LCM relation
For any two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b
Check with 96 and 404: 4 × 9696 = 38,784 = 96 × 404. ✓
This lets you find one when you know the other. If HCF(306, 657) = 9, then LCM = (306 × 657) ÷ 9 = 22,338.
Be careful: this relation is true for two numbers, not for three. For 6, 72 and 120, HCF × LCM = 2160, but the product of the numbers is 51,840.
Word problems
HCF answers "the largest size that divides several things exactly": the largest tile for a floor, the greatest number of equal groups.
LCM answers "when will events line up again": bells ringing together, runners meeting at the start.
Example. Two runners complete a round of a track in 18 minutes and 12 minutes. If they start together, they next meet at the starting point after LCM(18, 12) = 36 minutes.
Irrational numbers
A number is rational if it can be written as p/q with p and q integers and q ≠ 0. Numbers that cannot be written this way are irrational: √2, √3, √5, π and so on.
A useful fact (from the Fundamental Theorem): if a prime p divides a², then p divides a.
Proof that √2 is irrational
This is a proof by contradiction: assume the opposite and show it leads to something impossible.
Suppose √2 is rational. Then √2 = a/b, where a and b are integers with no common factor other than 1 (the fraction is in lowest terms) and b ≠ 0.
Squaring, 2 = a²/b², so a² = 2b². Hence 2 divides a², and therefore 2 divides a.
Write a = 2c. Then 4c² = 2b², so b² = 2c². Hence 2 divides b², and therefore 2 divides b.
So 2 divides both a and b, contradicting the assumption that they have no common factor.
Therefore √2 is irrational. The same method proves √3 and √5 are irrational.
Combinations of rational and irrational numbers
rational + irrational = irrational (e.g. 5 + √3)
non-zero rational × irrational = irrational (e.g. 2√5)
Example. Prove 3 + 2√5 is irrational. Suppose 3 + 2√5 = r, a rational number. Then √5 = (r − 3)/2, which would be rational. That contradicts the irrationality of √5. ∎
But irrational + irrational can be rational: (2 + √3) + (2 − √3) = 4.
Summary
Every composite number has a unique prime factorisation.
HCF uses the smallest powers of common primes; LCM uses the greatest powers of all primes.
HCF × LCM = product of the numbers, for two numbers only.
√2, √3, √5 are irrational, proved by contradiction.