#Numbers-7

(7^6n - 6^6n) , where n is an integer >0 ,is divisible by?

a)13  b)127 c)559 d)more than one of the above




Solution:-


Let n=1,

7^6- 6^6

we can break it to difference of two squares

(7^3)^2-(6^3)^2

(343 - 216 )(343+ 216)

So it can be divisible by 127 and 559 

When you get both the options, d is definitely the answer
but lets check for the first one also.

(7^2)^3-(6^2)^3 i.e (a-b)(a^2 +ab +b^2)...Difference of cubes

(49-36)(.....)
13(...)

One of the factors become 13..so it is divisible by 13 also

OR

Rule:-

(a^n - b^n) is always divisible by (a+b),if n is an Even Number.

Now,

6n is even

7+6=13,Hence the number is divisible by 13