Question

How many integer n exist so that \[ \frac{n}{100-n} \] is also an integer?

Answer 1

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?

Answer 2

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

Answer 3

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 \[(k+1)|100\]

Answer 4

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

Answer 5

(k+1) divides 100.

Answer 6

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

Answer 7

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

Answer 8

OK, thanks or that