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…') |
|||
(3 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | (a) Find all positive integers <math>n</math> for which <math>2^n-1</math> is divisible by <math>7</math>. | + | '''(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</math> for which <math>2^n+1</math> is divisible by <math>7</math>. | + | '''(b)''' Prove that there is no positive integer <math>n</math> for which <math>2^n+1</math> is divisible by <math>7</math>. |
== 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. | ||
+ | |||
+ | '''(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>. | ||
+ | |||
+ | == See Also == | ||
+ | {{IMO box|year=1964|before=First question|num-a=2}} |
Revision as of 12:45, 29 January 2021
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 .
See Also
1964 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |