Find the greatest 3 – digit number which when divided by 75, 45 and 60 leaves:
(A). No remainder
(B). The remainder 4 each case

Hint: First we will write all the prime factors of the given three numbers and then we will multiply them to get the number and then if it’s not the greatest 3 – digit then by multiplying with some integers to make it the greatest. And for the second case we just have to add 4 to the answer of the previous part to get the remainder 4 and hence that will be our final answer.

Complete step-by-step solution -
Let’s start our solution by writing the numbers in prime factor form.
Now let’s write the prime factors of the number:
$\begin{align}
  & 75=5\times 5\times 3 \\
 & 45=5\times 3\times 3 \\
 & 60=5\times 2\times 3\times 2 \\
\end{align}$
As we can see that the number has to be divisible by 75 so, we need two 5 and one 3,
$5\times 5\times 3$
Now for 45 we need one 5 and two 3, but we already have one 5 and one 3 in $5\times 5\times 3$ so, we will multiply our number with just 3.
After multiplying we get,
$5\times 5\times 3\times 3$
Now for the number to be divisible by 60 we already have 5 and 3 in $5\times 5\times 3\times 3$ , so we just need two 2.
After multiplying $5\times 5\times 3\times 3$ by two 2 we get,
$\begin{align}
  & 5\times 5\times 3\times 3\times 2\times 2 \\
 & =900 \\
\end{align}$
Now 900 is the smallest number that gives no remainder when divided by 75, 45, 60.
But it is also the largest 3 – digit number that gives no remainder when divided by 75, 45, 60.
Hence for (a) 900 is the correct answer.

Now we will proceed to part (b).
For the part (b) we will add 4 to 900 which gives 904.
And we know that 900 is divisible by 75, 45, 60 so, 904 will give us the required remainder 4 when divided by each number.
Hence, 904 is the correct answer for (b).

Note: One thing that we need to understand is how the smallest number that gives no remainder when divided by 75, 45, 60 is also the largest 3 – digit number that gives no remainder when divided by 75, 45, 60. Because when we multiply 900 by any integer > 1, then we get a 4 – digit number so 900 is largest as per the given condition.