SOLUTION: a,b,c,d are positive integers satisfying (a*b)+(b*c)+(c*d)+(d*a)=36 how many possible values of a+c are there?

Algebra ->  Average -> SOLUTION: a,b,c,d are positive integers satisfying (a*b)+(b*c)+(c*d)+(d*a)=36 how many possible values of a+c are there?      Log On


   

Question 1147347: a,b,c,d are positive integers satisfying (a*b)+(b*c)+(c*d)+(d*a)=36 how many possible values of a+c are there?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
(a*b)+(b*c)+(c*d)+(d*a)=36
We factor:
b(a+c)+d(c+a) = 36
b(a+c)+d(a+c) = 36
   (a+c)(b+d) = 36

a+c and b+d are divisors of 36, which are 1,2,3,4,6,9,12,18,36.

every factor of 36 except 1 and 36, is the sum of two factors of 36.

 2=1+1
 3=1+2
 4=1+3 =2+2
 6=2+4 =3+3
 9=3+6
12=3+9 =6+6
18=6+12=9+9

So there are 7 possible values of a+c.

Edwin