What Is The Least Common Multiple Of 15 And 9?

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Introduction

In mathematics, the least common multiple, commonly known as LCM, is the smallest positive integer that is divisible by two or more given numbers without a remainder. It is an important concept that is used in solving various problems in arithmetic and algebra. In this article, we will discuss the least common multiple of 15 and 9 and how to find it.

Method 1: Listing Multiples

One way to find the LCM of 15 and 9 is by listing their multiples and finding the smallest multiple that they have in common. To do this, we will first list the multiples of 15 and 9 as follows: 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, 915, 930, 945, 960, 975, 990, 1005, 1020, 1035, 1050 Multiples of 9: 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

Finding the Common Multiple

From the list above, we can see that the smallest multiple that 15 and 9 have in common is 45. Therefore, the LCM of 15 and 9 is 45.

Method 2: Prime Factorization

Another way to find the LCM of 15 and 9 is by using prime factorization. To do this, we will first factorize 15 and 9 into their prime factors as follows: 15 = 3 x 5 9 = 3 x 3

Finding the LCM

Next, we will take the highest power of each prime factor that appears in either factorization and multiply them together to get the LCM. In this case, the prime factors are 3 and 5, and the highest power of 3 is 3 x 3, which appears in the factorization of 9. The highest power of 5 is 5, which appears in the factorization of 15. Therefore, the LCM of 15 and 9 is 3 x 3 x 5 = 45.

Conclusion

In conclusion, the LCM of 15 and 9 is 45. We can find the LCM by listing the multiples of the two numbers and finding the smallest multiple they have in common or by using prime factorization. It is an important concept that is used in various mathematical problems, and understanding how to find it is crucial in solving these problems.