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

Question

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

Responses:

  • 900
  • 900
  • 15
  • 4545
  • 450
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Solution

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: LCM(a,b,c)=abcGCD(a,b,c) \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:

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

    • 75,150,225,300,375,450,525,600,675,750,825,900,75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, \dots
  • Multiples of 90:

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

Now, we can see:

  • The common multiples from above are 450450, 900900, etc.

4. ### Verify and Summarize

The smallest common multiple among the discovered common multiples is 450450 and the next one is 900900. Therefore, the greatest common multiple of the numbers is:

Final Answer

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

This problem has been solved

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