Aptitude - Problems on H.C.F and L.C.M - Discussion

let the greatest number is H.

Now;

H*Quotient(Q1)+Reminder(R)= 43 -------> (1)
H*Quotient(Q2)+Reminder(R)= 91 -------> (2)
H*Quotient(Q3)+Reminder(R)= 183 -------> (3)
Now, (1)-(2), (3)-(2), (3)-(1)
H*(Q2-Q1)= 48,
H*(Q3-Q2)= 92,
H*(Q3-Q1)= 140.

So 48, 92 and 140 are completely divisible by H, that will be the HCF of these numbers.
So, the HCF of 48, 92 and 140 is 4.

Thanks everyone for explaining the answer.

Why do we do the difference of number to each other?

Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

Explanation: 43, 91, 183.
183 = (183-91) = 92.
91 = (183-43) = 140.
43 = (91-43) = 48.

92, 140, 48.

92 = 2*2*23.
140 = 2*2*35.
48 = 2*2*2*3*2.

Ans 4.

Is there anyone who will clearly explain why one has to take the difference of those numbers?

I mean what is the key note in the question seeing which I will solve the problem in this way?

Please explain someone.

The answer is 4 and how it is 4 is below,

We can represent any integer number in the form of: D*q + r.
Where D is divisor, q is quotient, r is the remainder.

So each number can be written accordingly:
43 = D*q1 + r1;
91 = D*q2 + r2;
183 = D*q3 + r3;

r1, r2 & r3 will be same in above three equations according to the question.
D is the value that we want to find out. which should be greatest.

On solving three equations we get:

D*(q2-q1)= (91-43)=48
D*(q3-q2)= (183-91)=92
D*(q3-q1)= (183-43)=140
It is obvious that q3>q2>q1.

For the greatest value of D that divide each equation we take the HCF of 48,92,140

THEREFORE ANSWER IS 4.

Keep on dividing all the three numbers by 2 so that reminder should be the same ie
2[43 91 183]
[ 21 45 91] remainder will be 1 for all the three.

Divide again by 2.
2[21 45 91]
10 22 45 remainders will be same ie 1.

So after this, if you try to divide all three again by 2 then you get a different value for the reminder.
And hence; 2 * 2 = 4.

@All.

They asked to divide by a number also we need to have the same remainder in all cases.

So, dividing by 4, we get remainder 3 in all cases.

4/43 = remainder is 3.
4/183 =remainder is 3.
4/91 = remainder is 3.
So that' why the answer is 4.

48 = 4 * 12
92 = 4 * 23
140 = 4 * 35.

1,2,3,5 prime numbers so 4 is the correct answer.