What is the greatest number that divides 1,031, 1,751 and 2,759, leaving the same remainder in each case?

Explanation

1. 01

   If each leaves the same remainder, their differences are exact multiples of the divisor.

2. 02

   Differences: $1751 - 1031 = 720$, $2759 - 1751 = 1008$, $2759 - 1031 = 1728$.

3. 03

   $\operatorname{HCF}(720, 1008, 1728) = 144$ ($720 = 144 \times 5$, $1008 = 144 \times 7$, $1728 = 144 \times 12$).

4. 04

   Check: each number leaves remainder 23 when divided by 144.