Exploring the Smallest Three-Digit Number Divisible by 9, 12, and 15

To find the smallest three-digit number that is evenly divisible by 9, 12, and 15, we first need to calculate the least common multiple (LCM) of these three numbers. This process involves understanding the prime factorization of each number and then determining the LCM based on the highest powers of the prime factors.

Prime Factorization

Let's start by breaking down the numbers 9, 12, and 15 into their prime factors:

9: ( 3^2 )

12: ( 2^2 times 3^1 )

15: ( 3^1 times 5^1 )

Determining the Least Common Multiple

To find the LCM, we take the highest power of each prime factor found in any of the three numbers:

For 2: ( 2^2 ) from 12 For 3: ( 3^2 ) from 9 (since ( 3^1 ) from 15 and ( 3^2 ) from 9 are considered, but we take the highest power, which is ( 3^2 )) For 5: ( 5^1 ) from 15

Calculating the LCM

Using the highest powers of the prime factors, we calculate the LCM:

[ LCM 2^2 times 3^2 times 5^1 4 times 9 times 5 ]

Breaking it down step by step:

First, ( 4 times 9 36 ) Then, ( 36 times 5 180 )

Thus, the LCM of 9, 12, and 15 is ( 180 ).

Smallest Three-Digit Number Divisible by 180

Now, we need to find the smallest three-digit number that is divisible by 180. The smallest three-digit number is 100. To find the smallest multiple of 180 that is at least 100, we can use the following approach:

We divide 100 by 180 and round up to the nearest integer:

[ frac{100}{180} approx 0.5556 ]

Rounding up to the nearest integer gives us 1. Then, we calculate:

[ 180 times 1 180 ]

Therefore, the smallest three-digit number that is evenly divisible by 9, 12, and 15 is 180.

Alternative Approaches

Let’s also explore this problem using a brute force approach in the J programming language:

a~./09 12 15/a.100 to 999180 360 540 720 900

From this, we see that there are five three-digit integers that are evenly divisible by 9, 12, and 15. The smallest of these is 180.

Conclusion and Further Exploration

This example is a typical LCM problem, which is often used to demonstrate the application of prime factorization. The problem could be more interesting if the question asked for the smallest four-digit number instead. In that case, we would multiply the LCM (180) by 2, 3, 4, etc., until we produce a four-digit number, such as ( 180 times 6 1080 ).

Civilization advances by extending the number of important operations which we can perform without thinking of them. — Alfred North Whitehead

This quote highlights the importance of understanding mathematical concepts and how they form the foundation of advanced operations in various fields. By mastering the calculation of LCM, we can solve complex problems with ease and efficiency.