Prove That a Number is Divisible by Another Number

Question

Prove that the value of the expression: (125^2+25^2)(5^2–1) is divisible by 39

Answer 1

125²+25² = 16250 = 13*5⁴*2

5²-1 = 24 = 3*2³

 

Therefore the product (125²+25²)(5²+1) is divisible by 39 (=13*3).

 

 

Answer 2

There are many ways to show this.  I am showing an easy one.

(125^2 + 25^2)(5^2 - 1) = ((25*5)^2 + 25^2)(5^2 - 1) = 25^2(5^2 + 1)(5+1)(5-1) =

25^2(26)(6)(4) =

25^2(13*2)(3*2)(2*2) =

25^2(13)(3)2^4

Thus, 25^2(13)(3)2^4 / 39 = 25^2(13)(3)2^4 / (13*3) = 25^2 * 2^4

Yes, it is divisible by 39.