Difference between revisions of "1998 IMO Problems/Problem 4"
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Determine all pairs <math>(a, b)</math> of positive integers such that <math>ab^{2} + b + 7</math> divides | Determine all pairs <math>(a, b)</math> of positive integers such that <math>ab^{2} + b + 7</math> divides | ||
<math>a^{2}b + a + b</math>. | <math>a^{2}b + a + b</math>. | ||
| + | |||
| + | ===Solution=== | ||
| + | We use the division algorithm to obtain <math>ab^2+b+7 \mid 7a-b^2</math> | ||
| + | Here <math>7a-b^2=0</math> is a solution of the original statement, possible when <math>a=7k^2</math> and <math>b=7k</math> where <math>k</math> is any natural number. This is easily verified. | ||
| + | |||
| + | Otherwise we obtain the inequality (by basic properties of divisiblity): | ||
| + | <math>7a-b^2 \geq ab^2+b+7 \implies 7a+7-ab^2-b^2 \geq b+14 \implies (7-b^2)(a+1) \geq b+14</math> | ||
| + | So <math>7-b^2 \geq 0 \implies b=1,2</math> | ||
| + | |||
| + | Testing for <math>b=1</math> | ||
Revision as of 06:38, 10 April 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 
