Challenge Problem - K power

[agr_ritesh]

For positive integers k and n, the "k-power remainder of n" is defined as r in the following equation:
n = kw + r, where w is the largest integer such that r is not negative. For instance, the 3-power remainder of 13 is 4, since 13 = 32 + 4. In terms of k and w, what is the largest possible value of r that satisfies the given conditions?

Can you elaborate the explanation on "where w is the largest integer such that r is not negative"

[stud.jatt]

The stipulation "where w is the largest integer such that r is not negative" is to ensure a unique value for w and r each instead of a set of values.

for e.g. if we want to find the 3 power remainder of 26 we can have the following expressions

26= 3^1 + 23

26= 3^2 + 17

26= 3^3 - 1

in all these expressions the value of r fits the traditional meaning of a remainder only when it is +ve and it is the smallest value out of all possible. Hence the stipulation that w should be the largest integer such that r > 0 which in this case is satisfied by

26= 3^2 + 17

As for your original question "In terms of k and w, what is the largest possible value of r that satisfies the given conditions?" the answer is

r = (k-1)(k^w) - 1

[agr_ritesh]

Thanks for the explanation

[jnelson0612]

Great!