LCM of 15 and 12

Finding the LCM (Least Common Multiple) of two numbers is a common problem encountered in mathematics. In this article, we will focus on finding the LCM of 15 and 12, two relatively small numbers.

We will explore three methods to find the LCM of 15 and 12: prime factorization, listing multiples, and division method. We will also provide examples and frequently asked questions related to the LCM of 15 and 12.

What is the LCM of 15 and 12?

The LCM of two or more numbers is the smallest positive integer that is a multiple of all the numbers. In this case, we are interested in finding the LCM of 15 and 12. To do this, we need to find the smallest positive integer that is a multiple of both 15 and 12. This integer is called the LCM of 15 and 12.

LCM of 15 and 12 by Prime Factorization

One way to find the LCM of 15 and 12 is by using the prime factorization method. To use this method, we need to break down each number into its prime factors. For 15, we can write it as 3 x 5. For 12, we can write it as 2 x 2 x 3. Next, we need to identify the common prime factors and uncommon prime factors between the two numbers.

The common prime factors between 15 and 12 are 3 and 2. However, 12 has two 2’s in its prime factorization, while 15 does not have any. Therefore, we need to include both 2’s in the LCM. The LCM of 15 and 12 is then calculated by multiplying all the common factors and the remaining uncommon factors. Thus, the LCM of 15 and 12 is 2 x 2 x 3 x 5, which equals 60.

LCM of 15 and 12 by Listing Multiples

Another method to find the LCM of 15 and 12 is by listing multiples. To use this method, we need to list the multiples of each number until we find a common multiple. For 15, the multiples are 15, 30, 45, 60, 75, 90, 105, 120, etc. For 12, the multiples are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, etc.

From the lists above, we can see that the first common multiple of 15 and 12 is 60. Therefore, the LCM of 15 and 12 is 60.

LCM of 15 and 12 by Division Method

The division method is another approach to finding the LCM of two numbers. To use this method, we need to write the two numbers side by side and divide them by the smallest prime factor that divides both numbers evenly. We then repeat the process until we can no longer divide both numbers. Finally, we multiply all the prime factors and the remaining numbers to find the LCM.

For 15 and 12, the process looks like this:

Write 15 and 12 side by side: 15 | 12
Divide by the smallest prime factor, which is 3: 5 | 4
Divide by the smallest prime factor, which is 2: 5 | 2 | 2
The remaining numbers are 5 and 2. Multiply all the prime factors and the remaining numbers: 2 x 2 x 3 x 5 = 60.
Therefore, the LCM of 15 and 12 is 60.

LCM of 15 and 12 Examples

Let’s take a look at some examples to help solidify our understanding of finding the LCM of 15 and 12.

Example 1:

Find the LCM of 15 and 12.

Using the prime factorization method, we found that the LCM of 15 and 12 is 2 x 2 x 3 x 5 = 60.

Example 2:

Find the LCM of 15 and 12 using the listing multiples method.

The multiples of 15 are: 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, 915, 930, 945, 960, 975, 990, 1005, 1020, 1035, 1050, 1065, 1080, 1095, 1110, 1125, 1140, 1155, 1170, 1185, 1200…

The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636, 648, 660, 672, 684, 696, 708, 720, 732, 744, 756, 768, 780, 792, 804, 816, 828, 840, 852, 864, 876, 888, 900, 912, 924, 936, 948, 960…

From the lists above, we can see that the first common multiple of 15 and 12 is 60. Therefore, the LCM of 15 and 12 is 60.