This sum confusing me!

7 x6 = x2 + 2 x3

So , 7 x6 - [ x2 ] - [ 2 x3 ] = { x2 } + { 2 x3 }

Similarly observing that , 21 y20 = y4 + 4 y5

We may write down the given equation as -

{ x2 } + { 2 x3 } + { y4 } + { 4 y5 } = 0

Note that { b } ≥ 0 for all " b " , so each of these 4 terms must be zero .

Since , { a } = 0 if and only if , " a " is an integer , hence for each of these terms to be zero , " x " must be a multiple of " 6 " , and " y " has to be a multiple of " 20 " .

Accordingly , the total no. of ordered pairs of ( x , y ) are 4 .

shadi kar le sum se...propose kab kar ra hai?

waise though i think ricky left this part "PURPOSELY"...coz it does not matter much for this problem but still for sake of completeness eliminate the cases like
1.{x/2}+{2x/3}>1
{y/4}+{4y/5}<1
2.
{x/2}+{2x/3}>1
{y/4}+{4y/5}>1
3.{x/2}+{2x/3}<1
{y/4}+{4y/5}>1

Clearly ALL OF THE ABOVE CASES GET REJECTED IF WE OBSERVE THE GIVEN EQUATION CAREFULLY!

and now the problem is COMPLETELY CRACKED!