A positive integer is 5 less than another positive integer. Adding their reciprocals yields a sum of 1/6. What are the two positive integers.

Answer 1

x=y-6

1/x +1/y= 1/6

1/(y-6) +1/y = 1/6 multiply both sides by 6y(y-6)

6y + 6y -36 = y^2 - 6y combine the y terms

y^2 -18y +36 = 0 y = 9+ or - .5sqr(18^2 -4(36))= 9+/-3sqr5

= about (y -2,2918)(y-15.7082) = 0 there are no such integers DNE Do Not Exist, just irrational numbers that are 6 apart with reciprocal sum = 1/6

try some integers

1 7 1+1/7 = 8/7 sum too large

2 8 1/2 + 1/8 = 5/8 still too large

3 9 too large

4 10 too large

...

7 13 1/13 too large close though

8 14 closer

9 15 1/9 + 1/15 = 15/135 +9/135 = 24/135 = 8/45 > 1/7=8/56 still too large

...

100 106 1/100 + 1/106 = too small