Determine the smallest k for all +ve n!

[SilverSprite]

Determine the smallest integer k such that is divisible by 1441 for all positive integers n.

[Kalkewl8ter]

Hey, you're stealing all the problems from Dr White's math sheets!!!

[bugzpodder]

hmm can you do it KalKewl8ter? this one proves to be somewhat of a challenge. also u know that Dr. White 'stole' these problems from Euclid and Decartes contest?

[bugzpodder]

I hate you SS! you give me a question that has no solution and expect me to find a solution! I proved that in 5 minutes but apparently i wasn't 'confident enough' in my answers and kept searching for an answer for the next 1 and half hour with of course no luck, even using programming!!!

[Kalkewl8ter]

I agree with bugz, although I was almost positive that I had seen someone come up with a solution that led to an answer before...?

[bugzpodder]

i better provide a solution to back me up:

apparently if a+bz=cz (Rule X)
a=z(c-b)
then a must be divisible by z, if a,b,c,z are integers

all letters below are [positive] integers

60ⁿ+k*71ⁿ=1441x

60ⁿ⁺¹+k*71ⁿ⁺¹=1441y

ok second equation can be written as:

60*60ⁿ+71*k*71ⁿ=1441y

60*60ⁿ+60*k*71ⁿ+11*k*71ⁿ=1441y

60(60ⁿ+k*71ⁿ)+11*k*71ⁿ=1441y

60(1441x)+11*k*71ⁿ=1441y

By rule X,

11*k*71ⁿ=1441y' (where y' is also an integer)

k*71ⁿ=131y'

apparently since 131 is not divisible by 71, k must be divisible by 131

and we have:

60ⁿ+k*71ⁿ=1441x

when n=1

60+71k=1441x

60+71*(131y')=131*11*x

by Rule X,

60 has to be divisible by 131 which apparently is not, therefore no solution

[SilverSprite]

Kalkewl8ter, I dont know WHAT your talking about!