Concept 1.2
HCF and LCM from Prime Factors
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
If two numbers are written as products of primes, their HCF is the product of the smallest power of each common prime factor, and their LCM is the product of the greatest power of each prime factor involved.
The intuition
12 = 2² × 3 and 18 = 2 × 3². The HCF has to fit inside both numbers, so it can only take as many 2s and 3s as the poorer of the two has: 2 × 3 = 6.
The LCM has to contain both numbers, so it takes as many of each prime as the richer one has: 2² × 3² = 36. One question per prime: who has fewer, who has more?
The attention
Look at the power of each prime: HCF takes the smaller, LCM the larger.
12 = 2² × 3¹ 18 = 2¹ × 3²
Look hereNot here
- Look here
- The power of each prime. Compare the powers prime by prime: 2² against 2¹, 3¹ against 3². The HCF keeps the smaller of each, the LCM the larger.
- Not here
- The numbers themselves. Listing factors or multiples of 12 and 18 works for small numbers but hides the structure. For bigger numbers it becomes guesswork.
- What it teaches
- Break one big comparison into small, separate ones: one prime at a time.