PHP practice for RMO #1

Let $$n \geq 3$$ be an odd number.Show that there is a number in the set,$$\{2^1-1,2^2-1,...2^{n-1}-1\}$$,which is divisible by $$n$$.

Note by Adarsh Kumar
2 years, 8 months ago

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The question is easy. My proof: If say some of the numbers is divisible by n then we are done. If no then there must be two numbers which must be congruent to the same number x(mod n). Let those two numbers be (2^i-1) and (2^j-1). Then subtract the smaller from the larger and factorize. WLOG let i>j. Then u get 2^j common outside and inside the bracket u get (2^(i-j)-1). Now since n is odd, gcd(n,2^j)=1. So n divides (2^(i-j)-1) and note that this quantity lies in the set and hence you are done :)

Sorry for not using Latex as I am in a hurry right now !

- 2 years, 7 months ago

If I can ask,are you a RMO participant?

- 2 years, 7 months ago

Seems like I have got a tough competitor from my state ;)

- 2 years, 7 months ago

Yes

- 2 years, 7 months ago

This year?

- 2 years, 7 months ago

Yes

- 2 years, 7 months ago

Which class?

- 2 years, 7 months ago

This is my last year I am in 11th

- 2 years, 7 months ago

Ohh Well,best of luck!

- 2 years, 7 months ago

That is a nice solution and no problem!

- 2 years, 7 months ago