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Question

The greatest number which when divided by 43 , 91 and 183 leaves the same remainder is......( Please explain me method in detail)


Solution

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

HCF (48, 92 and 140)

As

48 = 2 x 2 x 2 x 2 x 3,

92 = 2 x 2 x 23,

140 = 2 x 2 x 5 x 7

HCF = 2 x 2 = 4.

And is the required number

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