## henpen 2 years ago How many integer n exist so that $\frac{n}{100-n}$ is also an integer?

1. henpen

What is the general strategy to be adopted for 'How many integer n exist so that f(n) is also an integer?', other than bursts of insight?

i think, trial error is the best way for this case

3. sirm3d

suppose that for some integer k, $k=\frac{ n }{ 100-n }$ rearranging the equation we get $n=\frac{ 100k }{ k+1 }$ and since k and k+1 are relatively prime, we conclude that $(k+1)|100$

4. henpen

What does the notation (k+1)|100 mean?

5. sirm3d

(k+1) divides 100.

6. sirm3d

when k = 0, n = 0. when k = 1, n = 50 when k = 3, n = 75

7. sirm3d

there are more values for k, but the solution set is finite.

8. henpen

OK, thanks or that