abhi750 wrote:

The answer is 5..

Guys, I do not understand how vcbabu solved it.. Neither do I know Eulers theorm or coprimes..

It is pretty straightforward..

5^3 = 125 .. and when 125 is divided by 63, the remainder is -1

=> 5^37/63

=> (5^36 * 5)/63

pls note that 5^36 will yield a remainder of 1 as it can be written as 5^(3*12)

so final remainder is 1 * 5 = 5

I didn't pick pen for this answer.. May be you have a better way to do it.. but I think this works as well..

This is actually a very clever solution which is actually a variation of the use of Euler's theorem. Forget about the fancy names -- in plain words, basically, if A and B do not have common factors (i.e., they are "coprime") then there exists an exponent n for which A^n / B will have remainder of 1.

For this problem, that number is 6 (the computation of which is beyond the scope of this forum). Hence if 5^6 mod 63 = 1 then 5^(6*6) mod 63 also has mod 1 and, similar to your reasoning, 5^(6*6 + 1) mod 63 = 5. You can determine that 5^6 mod 63 = 1, but that is quite difficult considering 125^2 is a five-figure number that you have to divide by 63.

Your solution is clever because most people (including me) are NOT taught to think of remainders in terms of negative numbers -- in fact, you are not guaranteed to find a remainder of -1 (though you will always find one of +1) and this works ONLY if the numbers do not have a common factor.

I still think this is too hard of a problem for the GMAT (need to accept concept of "negative remainder", need to apply some guesswork to find the exponent that yilds a remainder of -1 (which is not guaranteed) and understand that the remainder will alternate between -1 (or 62) and 1 for increasing multiples of 3 in the exponent), but perhaps i was wrong .

Whatever the case, kudos to you for solving this in a clever and straight forwad manner. Nice job.

_________________

Best,

AkamaiBrah

Former Senior Instructor, Manhattan GMAT and VeritasPrep

Vice President, Midtown NYC Investment Bank, Structured Finance IT

MFE, Haas School of Business, UC Berkeley, Class of 2005

MBA, Anderson School of Management, UCLA, Class of 1993