A quicker procedure

There is a procedure that finds the HCF and the LCM together, in one small table. It is close to the division method for prime factorisation, but run on two numbers side by side.

Write the two numbers in a row. Divide BOTH by a common prime factor, writing that prime on the left and the two quotients in the next row. Repeat. Take care: at every step the divisor must go into both numbers. Stop when the two numbers left share no common factor other than 1.

For 84 and 180: divide by 2 to get 42 and 90; by 2 again to get 21 and 45; by 3 to get 7 and 15. Now 7 and 15 share nothing, so stop.

The HCF is the product of the divisors down the LEFT: 2 × 2 × 3 = 12. That is right, because those are exactly the primes common to both, each taken as many times as both could supply.

Two more. For 300 and 150: divide by 2 (150, 75), by 5 (30, 15), by 5 (6, 3), by 3 (2, 1). HCF = 2 × 5 × 5 × 3 = 150. For 630 and 770: divide by 2 (315, 385), by 5 (63, 77), by 7 (9, 11). HCF = 2 × 5 × 7 = 70.

Now the LCM, from the same table. Multiply the divisors down the left by the two numbers left at the BOTTOM. For 300 and 150: 2 × 5 × 5 × 3 × 2 × 1 = 300. For 630 and 770: 2 × 5 × 7 × 3 × 3 × 11 = 6930. Why does that work? Because the product contains all the common primes once, plus everything each number has left over — which is exactly the LCM rule of taking each prime the maximum number of times.

Guna offers a shortcut: 'For 300 and 150, I can first divide both directly by 50.' That gives 6 and 3, then dividing by 3 gives 2 and 1, so the HCF is 50 × 3 = 150. It works — the divisor need not be prime, as long as it is a common factor — and it is faster when you can spot a large common factor.