indices
posted by rakesh .
Find a common factor of (527 to the power 379 + 473 to the power 379) and (527 to the power 166  473 to the power 166)

did you know that for
x^n + y^n, x + y is a factor if n is odd and for
x^n  y^n, xy is a factor in n is even
e.g. try some small numbers that fit into your calculator for testing
5^7 + 11^7 = 19565296 and
19565296 divides by 16 evenly 1222831 times
13^6  5^6 = 4811184 and
4811184 divides by 8 evenly 601398 times
I will use this property
527^{379} + 473^{379}
= (527+473)(some big number)
= 1000(some big number)
527^{166}  473^{166}
= (527473)(some other big number)
= 54(some other big number)
so the first one divides by 1000, the other by 54,
so a common factor is 2
note it said to find "a common" factor, not the highest common factor.
There may be others