The Lcm Of 9 And 12

What is the lowest common multiple of 12 and 9? [Solved] from www.cuemath.com

Introduction

When it comes to mathematics, one of the fundamental concepts is the Least Common Multiple (LCM). The LCM is the smallest number that is divisible by two or more given numbers. It is a crucial concept in mathematics, especially in algebra and geometry, as it helps solve problems that involve fractions, ratios, and percentages. In this article, we will discuss the LCM of 9 and 12, its definition, and how to find it. We will also provide examples and step-by-step solutions to make it easier to understand.

Definition of LCM

The LCM of two or more numbers is the smallest positive integer that is divisible by all of them. In other words, it is the lowest common multiple of a given set of numbers. To find the LCM, we need to identify the multiples of each number and find the smallest one that is common to all of them.

Multiples of 9 and 12

To find the LCM of 9 and 12, we first need to identify their multiples. A multiple of a number is the product of that number and any positive integer. For example, the multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477, 486, 495, 504, 513, 522, 531, 540, 549, 558, 567, 576, 585, 594, 603, 612, 621, 630, 639, 648, 657, 666, 675, 684, 693, 702, 711, 720, 729, 738, 747, 756, 765, 774, 783, 792, 801, 810, 819, 828, 837, 846, 855, 864, 873, 882, 891, 900. Similarly, 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.

Common Multiples of 9 and 12

Now that we have identified the multiples of 9 and 12, we need to find the common multiples. These are the numbers that appear in both lists. The common multiples of 9 and 12 are: 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540, 576, 612, 648, 684, 720, 756, 792, 828, 864, 900.

Finding the LCM

To find the LCM of 9 and 12, we need to identify the smallest number that is common to both lists. In this case, the LCM is 36. This means that 36 is the smallest number that is a multiple of both 9 and 12.

Examples

Let us look at some examples to understand how to find the LCM of 9 and 12 using the above method. Example 1: Find the LCM of 9 and 12. Solution: The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477, 486, 495, 504, 513, 522, 531, 540, 549, 558, 567, 576, 585, 594, 603, 612, 621, 630, 639, 648, 657, 666, 675, 684, 693, 702, 711, 720, 729, 738, 747, 756, 765, 774, 783, 792, 801, 810, 819, 828, 837, 846, 855, 864, 873, 882, 891, 900. 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. The common multiples of 9 and 12 are 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540, 576, 612, 648, 684, 720, 756, 792, 828, 864, 900. Therefore, the LCM of 9 and 12 is 36. Example 2: Find the LCM of 9 and 12 using the prime factorization method. Solution: We can also find the LCM of 9 and 12 using the prime factorization method. To do this, we need to find the prime factors of each number and then multiply the highest powers of each factor together. The prime factors of 9 are 3 x 3, and the prime factors of 12 are 2 x 2 x 3. To find the LCM, we need to multiply the highest powers of each factor together. The highest power of 2 is 2 x 2 = 4, and the highest power of 3 is 3. Therefore, the LCM of 9 and 12 is 4 x 3 = 12.

Conclusion

The LCM of 9 and 12 is 36. We can find the LCM by identifying the multiples of each number and finding the smallest one that is common to both. We can also find the LCM using the prime factorization method by finding the prime factors of each number and multiplying the highest powers together. The LCM is a crucial concept in mathematics, and it helps solve problems that involve fractions, ratios, and percentages.