Difference between revisions of "1998 IMO Problems/Problem 4"
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Hence, the only solutions are: | Hence, the only solutions are: | ||
− | <math>(a,b) = (11, 1), (49,1), (7k^2, 7k</math> for all natural <math>k</math>. | + | <math>(a,b) = (11, 1), (49,1), (7k^2, 7k)</math> for all natural <math>k</math>. |
+ | |||
+ | Written by dabab_kebab | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=1998|num-b=3|num-a=5}} |
Latest revision as of 23:48, 18 November 2023
Determine all pairs of positive integers such that
divides
.
Solution
We use the division algorithm to obtain
Here
is a solution of the original statement, possible when
and
where
is any natural number. This is easily verified.
Otherwise we obtain the inequality (by basic properties of divisiblity):
So
Testing for we find that
Therefore,
, and we can easily check these.
Testing for and applying the division algorithm we find that
, having no solutions in natural
.
Hence, the only solutions are:
for all natural
.
Written by dabab_kebab
See Also
1998 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |