Factors & Multiples
Prime Factorization
Prime factorization means writing a number as a product of prime numbers. A factor tree helps: keep splitting a number into factors until only primes are left.
For 12: 12 = 2 × 6 = 2 × 2 × 3. So the prime factorization of 12 is 2 × 2 × 3.
- Prime factorization = a number written as a product of primes.
- Use a factor tree, splitting until only primes remain.
HCF & LCM by Prime Factorization
Write each number as a product of primes. The HCF is the product of the common prime factors; the LCM is the product of all prime factors, taking the highest power of each.
For 12 = 2² × 3 and 18 = 2 × 3²: HCF = 2 × 3 = 6 and LCM = 2² × 3² = 36.
- HCF = product of common prime factors.
- LCM = product of all primes, using the highest power of each.
Divisibility Rules
Quick checks: a number is divisible by 4 if its last two digits form a number divisible by 4, by 8 if its last three digits do, by 6 if it is divisible by both 2 and 3, by 9 if the digit sum is divisible by 9, and by 11 if the difference of alternate digit sums is 0 or a multiple of 11.
- By 4: last two digits; by 8: last three digits.
- By 6: divisible by 2 and 3; by 9: digit sum divisible by 9.