Key Concepts
- 1Fundamental Theorem of Arithmetic
- 2HCF × LCM = ?
- 3Terminating decimal condition
- 4Euclid's Division Lemma
- 5Irrational number
- 6Co-prime numbers
- 7HCF by prime factorisation
- 8LCM by prime factorisation
- 9Fundamental Theorem of Arithmetic
- 10HCF × LCM = ?
Important Formulas & Facts
Every composite number can be expressed as a product of primes uniquely (apart from order).
Product of the two numbers: HCF(a,b) × LCM(a,b) = a × b
Denominator (in simplest form) has only factors of 2 and/or 5.
a = bq + r where 0 ≤ r < b. Used repeatedly to find HCF.
A number that cannot be expressed as p/q (q ≠ 0). Examples: √2, √3, π
Two numbers whose HCF is 1. Example: 7 and 11.
Take the lowest power of each common prime factor.
Take the highest power of each prime factor appearing in either number.
Every composite number can be expressed as a product of primes uniquely (apart from order).
Product of the two numbers: HCF(a,b) × LCM(a,b) = a × b
Must-Know Questions
Q1The HCF of 96 and 404 is:
96 = 2⁵ × 3, 404 = 2² × 101. HCF = 2² = 4.
Q2If HCF(306, 657) = 9, find LCM(306, 657).
HCF × LCM = Product. 9 × LCM = 306 × 657 = 201042. LCM = 22338.
Q3Prove that √2 is irrational.
Assume √2 = p/q where p, q are coprime integers. Then 2 = p²/q², so p² = 2q². This means p² is even, so p is even. Let p = 2k. Then 4k² = 2q², giving q² = 2k², so q is also even. This contradicts p, q being coprime. Hence √2 is irrational.
Q4The decimal expansion of 17/8 terminates after how many decimal places?
17/8 = 17/2³. Denominator has only factor 2. 17/8 = 2.125 → 3 decimal places.
Q5Find the LCM of 12, 15 and 21 by prime factorisation.
12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7. LCM = 2² × 3 × 5 × 7 = 420.
Practice Real Numbers
Reinforce what you just revised with practice questions