In finding the HCF of two numbers by division method, the quotients are 1, 5, and 8, respectively, and the last divisor is 57. What is the LCM of the two numbers?

Correct option is D

Given:

Quotients during the HCF calculation using the division method are: 1, 5, and 8

The last divisor (which is the HCF) is 57

Concept Used:

The division method of HCF follows the Euclidean Algorithm, where:

If we perform divisions like this:

N1 = N2 × q1 + r1 (Step 1)

N2 = r1 × q2 + r2(Step 2)

r1 = r2 × q3 + 0(Step 3)

Here, q1, q2, q3  are the given quotients

The last divisor  = r2, which is also the HCF

We start backward from the last step to determine the original numbers  

LCM × HCF = N1 × N2 , where N1 and N2 are two numbers

Solution:

From Step 3:

r1 = 57 × 8 = 456

From Step 2:

N2 = 456 × 5 + 57 = 2280 + 57 = 2337

From Step 1:

N1 = 2337 × 1 + 456 = 2337 + 456 = 2793

So, the two original numbers are:

N1 = 2793, N2 = 2337 and HCF = 57

Now,  

LCM × HCF = N1 × N2

LCM × 57 =2793 × 2337

LCM = $$\frac{2793 \times 2337}{57} = 49 \times 2337$$ = 114513  

Alternate Solution:
r1 = 57 × 8 = 456 
N2 = 456 × 5 + 57 = 2280 + 57 = 2337
N1 = 2337 × 1 + 456 = 2337 + 456 = 2793 
LCM × HCF = N1 × N2

LCM × 57 =2793 × 2337

LCM = $$\frac{2793 \times 2337}{57} = 49 \times 2337$$ = 114513