Final answer:
To find the greatest 5-digit number that satisfies the given conditions, find the LCM of the divisors and set up simultaneous equations to find a suitable value of k. The highest 5-digit number that satisfies given conditions is 99543.
Step-by-step explanation:
To find the greatest 5-digit number that satisfies the given conditions, we can start by finding the least common multiple (LCM) of the numbers 9, 12, 24, and 45.
The LCM of these numbers is 360.
When a number is divided by its LCM, the remainder is the same as when it is divided by each individual number.
So, we need to find the highest multiple of 360 that leaves remainders of 3, 6, 12, and 39 when divided by 9, 12, 24, and 45 respectively.
A number that leaves a remainder of 3 when divided by 9 is of the form 9k + 3.
Similarly, a number that leaves a remainder of 6 when divided by 12 is of the form 12k + 6.
Using these forms, we can set up simultaneous equations to find a suitable value of k that satisfies all the given conditions.
Hence, the highest 5-digit number that satisfies these conditions is 99543.