What is the greatest common multiple of 15, 75 and 90?Responses9009001515450450

To find the greatest common multiple (GCM) of the numbers 15, 75, and 90, we can follow these steps:

1. ### Break Down the Problem

We need to find the GCM of three numbers. The GCM is the smallest multiple that is common to all three numbers.

2. ### Relevant Concepts

The GCM can often be found using the prime factorization of each number or using the least common multiple (LCM) method, expressed as follows: $\text{LCM}(a, b, c) = \frac{|a \cdot b \cdot c|}{\text{GCD}(a, b, c)}$ However, for the explanation here, we'll focus on finding the direct multiples.

3. ### Analysis and Detail

Let's find the multiples of each number up to a reasonable limit:

• Multiples of 15:

  • $15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360, 375, 390, 405, 420, 435, 450, 465, 480, 495, 510, 525, 540, 555, 570, 585, 600, 615, 630, 645, 660, 675, 690, 705, 720, 735, 750, 765, 780, 795, 810, 825, 840, 855, 870, 885, 900, \dots$

• Multiples of 75:

  • $75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, \dots$

• Multiples of 90:

  • $90, 180, 270, 360, 450, 540, 630, 720, 810, 900, \dots$

Now, we can see:

• The common multiples from above are $450$, $900$, etc.

4. ### Verify and Summarize

The smallest common multiple among the discovered common multiples is $450$ and the next one is $900$. Therefore, the greatest common multiple of the numbers is:

Final Answer

$\text{The greatest common multiple of 15, 75 and 90 is } 900.$