# Difference between revisions of "1964 IMO Problems/Problem 1"

(Created page with '== Problem == (a) Find all positive integers <math>n</math> for which <math>2^n-1</math> is divisible by <math>7</math>. (b) Prove that there is no positive integer <math>n</mat…') |
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== Solution == | == Solution == | ||

+ | We see that <math>2^n</math> is equivalent to <math>2, 4,</math> and <math>1</math> <math>\pmod{7}</math> for <math>n</math> congruent to <math>1</math>, <math>2</math>, and <math>0</math> <math>\pmod{3}</math>, respectively. | ||

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+ | (a) From the statement above, only <math>n</math> divisible by <math>3</math> work. | ||

+ | |||

+ | (b) Again from the statement above, <math>2^n</math> can never be congruent to <math>-1</math> <math>\pmod{7}</math>, so there are no solutions for <math>n</math>. |

## Revision as of 19:29, 15 July 2009

## Problem

(a) Find all positive integers for which is divisible by .

(b) Prove that there is no positive integer for which is divisible by .

## Solution

We see that is equivalent to and for congruent to , , and , respectively.

(a) From the statement above, only divisible by work.

(b) Again from the statement above, can never be congruent to , so there are no solutions for .