Question

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

Answer 1

We have to find Greatest Factor

In this case , we have to find HCF with remainder ( no mention of remainder in question)

Step

Find the Differences of numbersGet the HCF ( that differences)

We have here 43 ,91 and 183

So differences are

183 - 91 = 92,

183 - 43 = 140,

91 - 43 = 48.

Now

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 4 is the required number.

There's an alternate method too:

Let the number be N, then we have -

43 ≡≡ 91 mod N

43 ≡≡ 183 mod N

and 183 ≡≡ 91 mod N

which means that N divides (91 - 43), (183 - 43) and (183 - 91)

=> N divides 48, 140 and 92

The greatest possible value of N is GCD of 48, 140 and 92, which is 4

Hope This Helps :)