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to get remainder $$$x \ (2 * x \lt a)$$$ you can do always do $$$a_i mod (a_i - x)$$$ as
$$$y = a_i - x$$$
$$$2y = 2a_i - 2x$$$
$$$a_i + (a_i - 2x) \gt a$$$
$$$2y \gt a$$$

Hi Divya,

Lets say x is the number and x % d = rem. So this means x = q*d + rem where r < d and hence max value of r can be min(d - 1, x - d).

When will r has it max value? When d - 1 = x - d. So 2d - 1 = x and hence d = (x + 1) / 2. So when we take d - 1 to be r we get r = (x + 1) / 2 - 1. Hope this helps!