In finding the HCF of two numbers by division method, the quotients are 4, 8…

The division (Euclidean) method finds the HCF of two numbers by repeated division: at each step, dividend = divisor times quotient plus remainder, and the divisor together with the remainder become the dividend and divisor of the next step; the process ends when the remainder becomes 0, and the divisor of that last step is the HCF. For any two numbers, their product always equals the product of their HCF and their LCM.

1. The quotients obtained were 4, 8 and 7 (first, second and third division respectively), and the last divisor was 66; since the last step leaves remainder 0, this last divisor is the HCF, so HCF = 66.

2. Reconstruct the earlier remainder: at the third step, the previous remainder equals the third quotient times the last divisor, i.e. 7 times 66 = 462.

3. Reconstruct the smaller of the two numbers: at the second step, the second quotient times 462 plus the last divisor gives the smaller number, i.e. (8 times 462) + 66 = 3696 + 66 = 3762.

4. Reconstruct the larger of the two numbers: at the first step, the first quotient times 3762 plus 462 gives the larger number, i.e. (4 times 3762) + 462 = 15048 + 462 = 15510.

5. Using the HCF-LCM product relation, LCM = (larger number times smaller number) divided by HCF = (15510 times 3762) divided by 66 = 58,348,620 divided by 66 = 884070.

Running the division method forward on 15510 and 3762 reproduces exactly the given data: 15510 = (4 times 3762) + 462, 3762 = (8 times 462) + 66, and 462 = (7 times 66) + 0, confirming HCF = 66; and 884070 times 66 = 58,348,620 = 15510 times 3762, confirming the LCM by an independent product check.