Sanchi, rather than solving I will give you a hint. Take 12 common in numerator and denominator so you are left with (2 * 24^59 + 5 * 60^23) / 7. This should be easy to solve – find the answer and multiply by 12.
Not quite, you’ve made a slight error!
You wrote:
24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem (2^60/7) => rem = 12*1 = 12
But actually
24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem [(12^59 * 2^60)/7] => rem = 12*3 = 36
Similarly the other also will give 36 and hence the overall remainder will turn out to be 72.
So if you are dividing by 13, just start the normal procedure and keep going till you reach either 1 or 2. If you reach one, your job is done, if you reach 12, treat it as -1 and continue (otherwise you would have to go twice as far to reach 1). Deal on a case-by-case basis!
CAT-holics 
This negative remainder concept really saves a lot of time !
find the remainder when 24^60 + 60^24 /84?
Sanchi, rather than solving I will give you a hint. Take 12 common in numerator and denominator so you are left with (2 * 24^59 + 5 * 60^23) / 7. This should be easy to solve – find the answer and multiply by 12.
regards
J
Answer should be 24, right?
24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem (2^60/7) => rem = 12*1 = 12
60^ 24 = 12^60 * 5^60 div by 12*7 => 12* Rem (5^60/7) => rem = 12* (-1)^20 = 12
Hence, (12+12) div by 84 gives the total remainder => Remainder = 24
Not quite, you’ve made a slight error!
You wrote:
24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem (2^60/7) => rem = 12*1 = 12
But actually
24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem [(12^59 * 2^60)/7] => rem = 12*3 = 36
Similarly the other also will give 36 and hence the overall remainder will turn out to be 72.
regards
J
Is the answer 1?
is the answer for Sanchi’s question 72?
How can to identify when to apply the “Applying Negative Remainders”. Say for example 23456^34567 divided by 13.
So if you are dividing by 13, just start the normal procedure and keep going till you reach either 1 or 2. If you reach one, your job is done, if you reach 12, treat it as -1 and continue (otherwise you would have to go twice as far to reach 1). Deal on a case-by-case basis!
regards
J