Log in to add a comment

## Answers

81^n - 64^n = 17 k ( where k is any number.)

The Brainliest Answer!

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

N = 81^n - 64^n , let us say n is a non-negative integer, 0,1,2,3....

N =

N = 9^{2n} - 8^{2n}

= (9^n)2 - (8^n)^2

= (9^n + 8^n) (9^n - 8^n)

= (3^{2n} + 2^{3n}) [ 3^{2n} - 2^{3n} ] there are minimum two factors.

n = 0, N = 1 - 1 =

= 1, N = 81 - 64 =

= 2 , N = 81^2 - 64^2 = (81+64) * (81 - 64) =

= 3, N = 81^3 - 64^3 = (81 - 64) * (81^2 + 81 * 64 + 64^2) =

= 4, N = 81^4 - 64^4 = (81-64)(81+64)(81^2+64^2) =

n = 5, N =

n=6, N = (81^3+64^3)(81-64)(81^2+81*64+64^2) =

n=7, N = 17 * 1086985055521

Always, A^n - B^n will have (A-B) as a factor. Hence the given expression has 17 as a factor. Perhaps there are other repeating factors for some values of n.

Also it seems to be multiple of odd integers. So it is always an odd integer.

N =

__(81-64) * (81^{n-1}+81^{n-2}*64+81^{n-3}*64^2+....+81^2*64^{n-3}+81*64^{n-2}+64^{n-1})__N = 9^{2n} - 8^{2n}

= (9^n)2 - (8^n)^2

= (9^n + 8^n) (9^n - 8^n)

= (3^{2n} + 2^{3n}) [ 3^{2n} - 2^{3n} ] there are minimum two factors.

n = 0, N = 1 - 1 =

**0**= 1, N = 81 - 64 =

**17**= 2 , N = 81^2 - 64^2 = (81+64) * (81 - 64) =

**29 * 5 * 17**= 3, N = 81^3 - 64^3 = (81 - 64) * (81^2 + 81 * 64 + 64^2) =

**17 * 73 * 7 * 31**= 4, N = 81^4 - 64^4 = (81-64)(81+64)(81^2+64^2) =

**17 * 29 * 5 * 10657**n = 5, N =

__17 * 11 * 41 * 491 * 641__n=6, N = (81^3+64^3)(81-64)(81^2+81*64+64^2) =

**17*5473*5*29*73* 7 * 31**n=7, N = 17 * 1086985055521

Always, A^n - B^n will have (A-B) as a factor. Hence the given expression has 17 as a factor. Perhaps there are other repeating factors for some values of n.

Also it seems to be multiple of odd integers. So it is always an odd integer.