Question

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

Answer 1

Given: 43, 91 and 183.

Step-wise calculation:

1. If a number d leaves the same remainder r when dividing 43, 91 and 183, then d divides the pairwise differences 91 – 43, 183 – 91 and 183 – 43.

Hence, d = HCF(91 – 43, 183 – 91, 183 – 43).

2. Compute the differences:

91 – 43 = 48

183 – 91 = 92

183 – 43 = 140

3. So, d = HCF(48, 92, 140).

4. Find HCF(48, 92):

92 = 48 × 1 + 44

48 = 44 × 1 + 4

44 = 4 × 11 + 0

⇒ HCF(48, 92) = 4.

5. Now HCF(4, 140):

140 is divisible by 4

⇒ HCF = 4

6. Check remainder: 

43 = 4 × 10 + 3

91 = 4 × 22 + 3

183 = 4 × 45 + 3

So, the common remainder is 3.

The greatest number is 4 (common remainder 3).