Q1.
Guna says “I found a better way to factorise to find HCF/LCM. This is faster than what was taught in class! For the numbers 300 and 150, I can first directly divide both numbers by 50. The HCF will be 50 × 3. The LCM will be 50 × 3 × 2 × 1”. Anshu also tried to remove the bigger common factors. “For 630 and 770, I will divide both numbers by 10 first. Now, I can divide them by 7. The HCF will be 10 × 7 = 70. The LCM will be 10 × 7 × 9 × 11 = 6930”. Can you see why this works?
Answer
It works because a step of the ladder never has to be a prime — any common factor will do.
Guna, on 300 and 150
50 | 300, 150 → 6, 3
3 | 6, 3 → 2, 1
HCF = 50 × 3 = 150
LCM = 50 × 3 × 2 × 1 = 300
50 | 300, 150 → 6, 3
3 | 6, 3 → 2, 1
HCF = 50 × 3 = 150
LCM = 50 × 3 × 2 × 1 = 300
Anshu, on 630 and 770
10 | 630, 770 → 63, 77
7 | 63, 77 → 9, 11
HCF = 10 × 7 = 70
LCM = 10 × 7 × 9 × 11 = 6930
10 | 630, 770 → 63, 77
7 | 63, 77 → 9, 11
HCF = 10 × 7 = 70
LCM = 10 × 7 × 9 × 11 = 6930
Both answers agree exactly with the slower prime-by-prime ladders on page 61.
Why it happens: dividing both numbers by 50 in one go is the same as dividing by 2, then by 5, then by 5 again — the three rows are merged into one. The left column still multiplies to the same thing, because 50 = 2 × 5 × 5. Nothing is skipped; the primes are only collected in bigger bundles. The one condition is that the divisor must be a common factor of both numbers, and that you keep going until the leftovers share nothing.
Tip: the shortcut is only safe if you finish the job. If Guna had stopped after dividing by 50, he would have got 50 as the HCF instead of 150 — because 6 and 3 still share a 3.