Please see this thread : http://gmatclub.com/forum/reminder-108378.html#p859740

Same methods are applicable here too.
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Anurag Mairal, Ph.D., MBA
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In the following explanation symbol a^b shoul be understood as A raised to the power and A/B is A divided by B.

1) I will try to explain a simple logic. Lets say if you were asked what is the remainder when 3/2, answer would be 1. Now lets say 3^2/2, in that case answer is 1*1, which is 1 again. Now let say if you were asked what was the remainder for 3^100/2, it would be 1*1*1*1.......100 times which is 1 again.
Till now what you have seen is the concept of positive remainder that is the divisor is less than the dividend

2) In the same way there is this concept called negative remainder, this would be applicable in the situation where divsor is less than the dividend. This brings me to the situation posed in the question asked. Let me give an example: what is the remainder when 2/3 it is 2 or (3-1)/3 or (3/3) - (1/3) or 0-1 or -1. So when 2 is divided by 3 i can say the remainder is 2 or -1. In the same way what would be the remainder when 2^2 / 3. It would be -1*-1, which gives me 1. In the same way lets say 2^2000/3 would be -1 multiplied 2000 times which gives me 1. This i can generalise as 2^even number when divided by 3 gives me -1 multipled even number of times resulting in 1. Had it been 2^odd number it gives me -1 multiplied odd number of times, that is negative remainder -1 or it could be translated to positive remainder 3-1 that is 2

If you understood what is being told try to solve these question
1) 4^123/3
2) 7^12345/8
the above questions are direct applications of positive remainder and negative remainder respectively

you got the answer right, but i saw the solution you gave to thevquestion posed. the question does nit translate to 3*25 it should 3 raised to the power 25.. And not 75. Change the question to 7^24/4 and find the answer to understand the mistake you made in the solution you posted

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