Step 1: Find the LCM of 48, 60 and 65 by prime factorisation:
$$48=2^4\times 3,\qquad 60=2^2\times 3\times 5,\qquad 65=5\times 13$$
Step 2: LCM $=$ product of highest powers of all prime factors:
$$\text{LCM}=2^4\times 3\times 5\times 13=16\times 195=3120$$
Step 3: Every number divisible by all three must be a multiple of 3120:
$$3120,\ 6240,\ 9360,\ 12480,\ \ldots$$
Step 4: The greatest such multiple less than 10,000 is $3120\times 3=9360$.
Answer: $9360$