Apply the Fundamental Theorem of Arithmetic to find HCF and LCM, and understand irrational numbers more deeply.
Every composite number can be expressed as a unique product of primes. This lets us find HCF and LCM using prime factorization.
Find the HCF and LCM of 12 and 18 using prime factorization.
12=2²×3, 18=2×3²; HCF=2×3=6; LCM=2²×3²=36
Find the HCF and LCM of 6 and 20.
6=2×3, 20=2²×5; HCF=2; LCM=2²×3×5=60
Find the HCF and LCM of 24 and 36 using prime factorization.
24=2³×3, 36=2²×3²; HCF=2²×3=12; LCM=2³×3²=72
Numbers like √2, √3, and √5 are proven irrational using proof by contradiction, assuming they're rational and reaching a contradiction.
Is √2 rational or irrational? Briefly explain.
Irrational; assuming √2=p/q (lowest terms) leads to both p and q being even, contradicting the lowest-terms assumption
Show that 5+√2 is irrational, given √2 is irrational.
If 5+√2 were rational, then √2=(5+√2)-5 would be rational too, a contradiction, so 5+√2 is irrational
Show that 2+√3 is irrational, given √3 is irrational.
If 2+√3 were rational, then √3=(2+√3)-2 would be rational too, a contradiction
1. Find the HCF and LCM of 15 and 20 using prime factorization.
2. Find the HCF and LCM of 8 and 12.
3. Is √5 rational or irrational?
4. Show that 3√2 is irrational, given √2 is irrational.
5. Find the HCF of 336 and 54 using prime factorization.
6. Find the HCF and LCM of 18 and 24 using prime factorization.
7. Find the HCF and LCM of 12 and 15.
8. Is √11 rational or irrational?
9. Show that 5√2 is irrational, given √2 is irrational.
10. Find the HCF of 96 and 404 using prime factorization.