Problem Analysis:
To find the greatest 4-digit number that is exactly divisible by 8, 10, and 12, we need to consider the multiples of the least common multiple (LCM) of these three numbers.

Least Common Multiple (LCM):
The LCM of 8, 10, and 12 can be found by finding the prime factorization of each number and taking the highest power of each prime factor that appears in any of the numbers:

Prime factorization of 8: 2^3
Prime factorization of 10: 2 * 5
Prime factorization of 12: 2^2 * 3

Taking the highest power of each prime factor, the LCM is calculated as:
LCM = 2^3 * 3 * 5 = 120

Finding the greatest 4-digit number:
Since we are looking for a 4-digit number, it should be greater than or equal to 1000 and less than or equal to 9999. We can start from the highest 4-digit number and check if it is divisible by 120.

Starting from 9999, we divide it by 120:
9999 ÷ 120 = 83 remainder 39

Since 39 is not divisible by 120, we subtract the remainder from 9999 and find the next multiple of 120:
9999 - 39 = 9960

9960 ÷ 120 = 83 remainder 0

Since the remainder is 0, 9960 is exactly divisible by 120.

Therefore, the greatest 4-digit number that is exactly divisible by 8, 10, and 12 is 9960.

Conclusion:
The greatest 4-digit number that is exactly divisible by 8, 10, and 12 is 9960. This number is obtained by finding the least common multiple (LCM) of 8, 10, and 12, which is 120, and then finding the highest multiple of 120 within the range of 4-digit numbers.

Lcm of 8,10,12 = 120



9999÷120=83.325 round off 83

83 × 120 = 9600