A number is divisible by 3 if the sum of its digits is divisible by 3. 

In 5763 sum of the digits is = $5+7+6+3=21$, which is exactly divisible by 3.

So, 5763 is divisible by 3.
A number is divisible by 4 if its last two digits are divisible by 4. The last two digits in 5763 are 63 which is not divisible by 4.
Hence 5763 is not divisible by 4

5763 is not divisible by 6, as it is an odd number. To be divisible by 6, the number should be divisible by both 3 and 2. Only even numbers are divisible by 2. 

A number is divisible by 9 if the sum of its digits is divisible by 9.

In 5763 sum of the digits is = $5+7+6+3=21$, which is not divisible by 9.

So, 5763 is not divisible by 9.
Hence 5763 is divisible by 3.