What is the largest four-digit number that is divisible by each of 15, 25 and 35?

To find the largest four-digit number divisible by $15$, $25$, and $35$, we must find the Least Common Multiple (LCM) of these three numbers.
1. Find the prime factorization of each number:

$$15=3\times 5$$

$$25=5^2$$

$$35=5\times 7$$

2. Calculate the LCM by taking the highest power of all prime factors involved:

$$\text{LCM}=3^1\times 5^2\times 7^1=3\times 25\times 7=525$$

3. The largest four-digit number is $9999$. Divide $9999$ by our LCM ($525$) to find the remainder:

$$9999÷525=19\text{ with a remainder of }24$$

Verification: $525\times 19=9975$
Remainder $=9999-9975=24$
4. Subtract the remainder from $9999$ to get the largest exactly divisible multiple:

$$\text{Required Number}=9999-24=9975$$

Therefore, $9975$ is the largest 4-digit number divisible by $15$, $25$, and $35$.