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Class 10 Real Numbers: cheat sheet
CBSE Mathematics. Every whole number above 1 is a product of primes in exactly one way; that gives HCF, LCM and proofs that numbers like √2 are irrational.
Key ideas
- Fundamental Theorem of Arithmetic: every composite number can be written as a product of primes in exactly one way, apart from the order of the factors.
- HCF: multiply the smallest power of each prime that appears in all the numbers.
- LCM: multiply the greatest power of each prime that appears in any of the numbers.
- For two numbers, HCF × LCM equals the product of the numbers. This does not hold for three numbers in general.
- A rational number can be written as p/q with integers p and q, q ≠ 0. An irrational number cannot.
- If a prime p divides a², then p divides a. This is the key step in proving √2, √3 and √5 irrational.
- Rational + irrational is irrational, and a non-zero rational × irrational is irrational.
Remember
| HCF and LCM of two numbers | HCF(a, b) × LCM(a, b) = a × b |
| Prime-power form | 360 = 2³ × 3² × 5 |
| From prime powers | HCF(2³ × 3, 2 × 3²) = 2 × 3 = 6; LCM = 2³ × 3² = 72 |
Watch out
- Using HCF × LCM = product for three numbers.
- Taking the greatest power for the HCF (it is the smallest).
- Writing 1 as a prime factor (1 is not prime).
Worked example
Find the HCF and LCM of 288 and 396 by prime factorisation.
Working: 288 = 2⁵ × 3²; 396 = 2² × 3² × 11; HCF = product of the lowest powers, LCM = the highest
Answer: HCF = 36, LCM = 3,168
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