I randomly pick an integer between 1 and 10 inclusive. What is the probablility that I choose a p such that there exists an integer q so that p and q satisfy the equation pq-4p-2q=2? Express your answer as a common fraction.

ANotSmartPerson
Oct 12, 2018

#2**+1 **

Thank you for trying, but the answer was incorrect :( Here is the solution I got back in case it helps.

We approach this problem by trying to find solutions to the equation pq-4p-2q=2 . To do this, we can use Simon's Favorite Factoring Trick and add 8 to both sides to get qp-4p-2q+8=10. This can be factored into

(p-2)(q-4)=10

We now can see that there are solutions only if p-2 divides 10 . Thus, there are 4 possible values of p between 1 and 10 inclusive (1,3,4,7) . It follows that the probability of picking such a p is 2/5.

Thanks again!

ANotSmartPerson
Oct 12, 2018

#1**+2 **

There are 3 numbers for p and corresponding 3 numbers for q that will satisfy your equation as follows:

p = (2 (q + 1))/(q - 4) and q must not =4

q = (4 p + 2)/(p - 2) and p must not=2

p =3 and q=14

p =4 and q=9

p =7 and q=6

Therefore the probability is:

**3/10 =30%**

Guest Oct 12, 2018

edited by
Guest
Oct 12, 2018

#2**+1 **

Best Answer

Thank you for trying, but the answer was incorrect :( Here is the solution I got back in case it helps.

We approach this problem by trying to find solutions to the equation pq-4p-2q=2 . To do this, we can use Simon's Favorite Factoring Trick and add 8 to both sides to get qp-4p-2q+8=10. This can be factored into

(p-2)(q-4)=10

We now can see that there are solutions only if p-2 divides 10 . Thus, there are 4 possible values of p between 1 and 10 inclusive (1,3,4,7) . It follows that the probability of picking such a p is 2/5.

Thanks again!

ANotSmartPerson
Oct 12, 2018