[24^50 - 15^50 ] / 13 =
[ 24^25 + 15^25 ] [ 24^25 - 15^25] / 13 =
[ (8 * 3)^25 + (5 * 3)^25 ] [ (8 * 3)^25 - (5 * 3)^25 ) / 13 =
[ 8^25 * 3^25 + 5^25 * 3^25 ] [ 8^25 * 3^25 - 5^25 * 3^25 ] / 13 =
[ 3^25 ( 8^25 + 5^25) ] [ 3^25 ( 8^25 - 5^25)] / 13
[ 3^50 ] [ 8^25 + 5^25 ] [ 8^25 - 5^25 ] / 13
Note that
(8^5) mod 13 = 8
And by a modular math property, since 5 is a factor of 25....then (8^25)mod 13 = 8
Also
(5^5) mod 13 = 5
And......using the same property......(5^25) mod 13 = (5^5) mod 13 = 5
So [ 8^25 + 5^25 ] /13 = [ 8^25 mod 13 + 5^25 mod 13 ] /13 =
[ 8mod 13 + 5 mod 13 ]/13 = [8 + 5] /13 = 13 /13 = 13 mod 13 = 0 = remainder
So
[ 3^50] * [ 8^25 + 5^25] * [ 8^25 - 5^25]
____________
13
Is evenly divisible by 13...so
(24*50 - 15^50) is evenly divisible by 13.....so.....the remainder = 0