Question

For what positive value n < 1000 is (n^2 - 75)/(5n + 56) also an integer?
How to solve this problem and what should I do in general when I get such problem?

Answer 1

hello thereq

Answer 2

(n^2 - 75)/(5n + 56)

Answer 3

Yes.

Answer 4

do you have any other forms of this equation i can look at that uve tried

Answer 5

Well, I don't know what to do, I tried to play with modulus, but it got nowhere, so no. I don't have anything else made.

Answer 6

k lemme try some stuff gimme a few mins

Answer 7

OK i suggest you start by synthetic division

Answer 8

yo i'm getting that there exists no such n

Answer 9

There exists one. of them.

Answer 10

oh wait yes

Answer 11

suppose n is even
then the numerator becomes odd and denominator becomes even => odd/even ->not integer

suppose n is odd
then the numerator becomes even and denominator becomes odd => even/odd->not integer(except when denominator=1or-1)

Answer 12

but for the denominator to become 1 n has to be negative but its given that n is positive

Answer 13

one more case could be that denominator becomes 0 but thats not possible cuz n^2 can't equal 75 because n in integer

Answer 14

i like that

Answer 15

however, solution is 241.

Answer 16

oh damn i just forgot that even/odd can be an integer
example 6/3

Answer 17

yes

Answer 18

so we know that n can't be even tho

Answer 19

yup

Answer 20

i gotts go eat food try playing with htis for now

Answer 21

1/25 * (5n-56 + 1261/(5n+56))

Answer 22

I solved it by brute force. Use Excel to plug the domain into the equation and inspect the range.

I initially tried to plot the equation, but get the resolution resolved fine enough.

Answer 23

so first of all 1261/that has to be a integer so

Answer 24

that shud narrow it down a lot

Answer 25

n=241 is when that goes to 1