Find the largest number that divides 155, 590, and 735 to leave the same remainder in each case.

Correct option is A

​$$\textbf{Let the required number be } N. \\\text{Since the same remainder is left when } 155, 590, \text{ and } 735 \text{ are divided by } N, \\\text{then } N \text{ must divide the differences of the numbers.} \\\text{Compute the pairwise differences:} \\590 - 155 = 435 \\735 - 590 = 145 \\735 - 155 = 580 \\\text{Now find } \gcd(435, 145, 580): \\\text{Prime factorization:} \\435 = 3 \times 5 \times 29 \\145 = 5 \times 29 \\580 = 2^2 \times 5 \times 29 \\\text{Common factors: } 5 \text{ and } 29 \\\Rightarrow \gcd(435, 145, 580) = 5 \times 29 = 145 \\\text{Hence, the required number is } \boxed{145}.$$​​