Question

hey guys, i saw this posted earlier, actually interested on how to solve it, líl help workin it out would be appreciated please?

Find the number of positive integers n less than 1000 such that 20^n - 13^n - 7^n is divisible by 309

Answer 1

so would u use logs, or is there a formula, or a simpler logical way?

Answer 2

I think its not by log because we can't take log inside the brackets
Log(20^n-13^n-7^n)

Answer 3

hmmm... ok, so what would u recommend going with?
thought maybe working out the primes of 309 (3 and 103)... then using that somehow...?

Answer 4

What is (13+7)^n
I think its first and last term will cancelled 13^n and 7^n

Answer 5

form what i remember tho: that's not a rule of exponentials, is it?

Answer 6

Isn't it binomial

Answer 7

well... for example:
if n=3
13^3 +7^3 = 2540
(13+7)^3 = 20^3 = 8000

Answer 8

No. What is binomial theorem

Answer 9

(13+7)^3=13^3+3(13^2)(7)+3(7^213)+7^3

Answer 10

ah... my bad... but they're like terms(no variables)...we cant add them in this scenario?

Answer 11

Because I want to cancel the (-13^n-7^n) part of question

Answer 12

i understand... but that still doesn't seem to math properly in my head... sorry :(

Answer 13

And we know 13^n will be first and 7^n will be last term of (13+7)^n

Answer 14

20^n- 13^n -7^n
=(13+7)^n- 13^n -7^n
=13^n + middle part of expansion+7^n -13^n -7^n
=middle part of expansion
In this way we are left with middle part of expansion
We have to find any relation between this part and 309

Answer 15

I'm interested to see how this resolves. Because the way I would do this would be to just write a program to tell me lol

Answer 16

Do you know. (a + b)^n = ??

Answer 17

yep, ur totally right, I'm with u now, sorry

Answer 18

Its not working ,I try it with pen and paper very confused

Answer 19

@ganeshie8 may help

Answer 20

i think @mukushla knows how to solve this.