Introduction:
In this problem, we need to find the smallest 5-digit number that is exactly divisible by 129. To solve this problem, we need to understand the concept of divisibility and how to find the LCM (Least Common Multiple) of two numbers.

Divisibility:
A number is said to be divisible by another number if it can be divided exactly by that number without leaving any remainder. For example, 10 is divisible by 2 because 10/2 = 5 with no remainder.

LCM:
The LCM of two or more numbers is the smallest number that is divisible by all of them. For example, the LCM of 3 and 4 is 12 because 12 is the smallest number that is divisible by both 3 and 4.

Solution:
To find the smallest 5-digit number that is exactly divisible by 129, we need to find the LCM of 129 and the smallest 5-digit number (10,000). We can do this by using the following steps:

Step 1: Prime factorize 129
129 = 3 x 43

Step 2: Prime factorize 10,000
10,000 = 2^4 x 5^4

Step 3: Find the LCM
To find the LCM, we need to take the highest power of each prime factor from both numbers. Therefore, the LCM of 129 and 10,000 is:
LCM = 2^4 x 3 x 5^4 x 43
LCM = 2,322,000

Step 4: Check if the LCM is a 5-digit number
The LCM we found is 2,322,000 which is a 7-digit number. Therefore, we need to find the next multiple of 2,322,000 that is a 5-digit number.

Step 5: Find the next multiple of 2,322,000 that is a 5-digit number
To find the next multiple of 2,322,000 that is a 5-digit number, we can divide 99,999 (the largest 5-digit number) by 2,322,000 and round up to the nearest whole number. This gives us:
Next multiple = ceil(99,999 / 2,322,000) x 2,322,000
Next multiple = 4 x 2,322,000
Next multiple = 9,288,000

Therefore, the smallest 5-digit number that is exactly divisible by 129 is 9,288,000.

Conclusion:
To find the smallest 5-digit number that is exactly divisible by 129, we need to find the LCM of 129 and the smallest 5-digit number (10,000) and then find the next multiple of the LCM that is a 5-digit number. The smallest 5-digit number that is exactly divisible by 129 is 9,288,000.

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