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Challenge Problem - K power

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agr_ritesh
 Post subject: Challenge Problem - K power
 Post Posted: Sat Dec 03, 2011 12:59 am 
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Students


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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"
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stud.jatt
 Post subject: Re: Challenge Problem - K power
 Post Posted: Sun Dec 04, 2011 12:54 pm 
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Students


Posts: 21
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
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agr_ritesh
 Post subject: Re: Challenge Problem - K power
 Post Posted: Sat Dec 10, 2011 11:44 pm 
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Thanks for the explanation
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jnelson0612
 Post subject: Re: Challenge Problem - K power
 Post Posted: Mon Dec 26, 2011 12:15 am 
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Great!

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Jamie Nelson
ManhattanGMAT Instructor
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