The greatest number of four digits which when divided by 3, 5, 7, 9leave remainders 1, 3, 5, 7 respectively is

Step 1: Find the least common multiple (LCM) of the divisors (3, 5, 7, 9). The LCM of these numbers is 315.

Step 2: The number we are looking for is of the form 315k + d, where k is a non-negative integer and d is the common difference of the remainders. In this case, d = 1 - 3 = 3 - 5 = 5 - 7 = -2.

Step 3: Since the number is a four-digit number, we need to find the maximum value of k such that 315k + d is a four-digit number. This means 315k - 2 < 10000. Solving this inequality gives k < 31.75.

Step 4: Since k must be an integer, the maximum value of k is 31. Substituting k = 31 into the equation 315k - 2 gives 315*31 - 2 = 9753.

So, the greatest number of four digits which when divided by 3, 5, 7, 9 leave remainders 1, 3, 5, 7 respectively is 9753.