Question

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)

Answer 1

did you know that for

x^n + y^n, x + y is a factor if n is odd and for
x^n - y^n, x-y 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

527³⁷⁹ + 473³⁷⁹
= (527+473)(some big number)
= 1000(some big number)

527¹⁶⁶ - 473¹⁶⁶
= (527-473)(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

Answer 2

(X^n - Y^n) is divisible by both ( x+a)& ( x - a ) if n is even,
Please be clear.