how do i prove the divisibility of an equation?

To prove that 5¹⁸ - 25⁸ is divisible by 120, write 25 as 5² and:

     5¹⁸ - 25⁸  =  5¹⁸ - (5²)^{8  =}  5¹⁸ - 5¹⁶

Now, factor out 5¹⁶:

     5¹⁸ - 5¹⁶   =    5¹⁶(5² - 1)    =    5¹⁶(25​ - 1)    =     5¹⁶(24) 

Since  120  =  24 x 5:

    5¹⁶(24)  /  (24 x 5)    =    5¹⁵

how do i prove that "5^18 - 25^8" is divisible by 120

For large numbers such as these, the easiest way is to do the Math on a calculator and find out that way:"5^18 - 25^8" is divisible by 120

5^18=3,814,697,265,625 - (25^8)=152,587,890,625=3,662,109,375,000 / 120=30,517,578,125.

This way, you don't have to guess.

5^18 - 25^8 
= 5^18 - 5^16 
= 5^16(5² – 1) 
= 5^16(24) 
= 24 × 5 × 5^15 
divide by 120 = 120 × 5^15/120 divisible 
= 5^15 

38^9 - 38^8 
= 38^8(38 –1) 
= 38^8 × 37 
divisible by 37 

5^18 - 25^8=5^18 - 5^16 mod 120 =0. It is divisible evenly by 120!!.