LCM of 63,70 and 77

Introduction

Finding the Least Common Multiple (LCM) is an important concept in math, especially when working with whole numbers. The LCM of a set of numbers is the smallest multiple that is evenly divisible by all the numbers in that set. In this article, we will look into how to find the LCM of 63, 70, and 77. By the end of this guide, you will have a clear understanding of how to calculate the LCM using different methods.

What is LCM?

The LCM, or Least Common Multiple, is the smallest multiple that two or more numbers share. For example, the LCM of 4 and 5 is 20, as 20 is the smallest number that both 4 and 5 divide evenly into. This concept is particularly useful when adding, subtracting, or comparing fractions, as having a common multiple can simplify these operations.

Ways to Calculate the LCM

There are several methods to calculate the LCM, including:

1. Division Method
2. Listing Method
3. Prime Factorization Method

Each method has its unique features, and we will explore each in detail.

Prime Factorization Method

The Prime Factorization Method involves breaking down each number into its prime factors and then using these factors to find the LCM.

Steps to calculate the LCM of 63, 70, and 77 using Prime Factorization:

1. Find the prime factors of each number.
   • 63 = 3 x 3 x 7
   • 70 = 2 x 5 x 7
   • 77 = 7 x 11
2. Identify all unique prime factors from the numbers.
   • 2, 3, 5, 7, 11
3. Take the highest power of each prime factor.
   • 2^1, 3^2, 5^1, 7^1, 11^1
4. Multiply these together to get the LCM.
   • LCM = 2 x 3^2 x 5 x 7 x 11 = 2 x 9 x 5 x 7 x 11 = 2 x 9 x 35 x 11 = 2 x 315 x 11 = 630 x 11 = 6930

Therefore, the LCM of 63, 70, and 77 is 6930.

Division Method

The Division Method involves dividing the numbers simultaneously by their common prime factors until only 1s are left.

Steps to calculate the LCM of 63, 70, and 77 using the Division Method:

1. Write the numbers in a row: 63, 70, 77.
2. Divide by the smallest prime number that can divide at least one of the numbers.
   • Divide by 2: 63, 35, 77
   • Divide by 3: 21, 35, 77
   • Divide by 3: 7, 35, 77
   • Divide by 5: 7, 7, 77
   • Divide by 7: 1, 1, 11
   • Divide by 11: 1, 1, 1
3. Multiply all the divisors: 2 x 3 x 3 x 5 x 7 x 11 = 6930

So, the LCM of 63, 70, and 77 is 6930.

Listing Method

The Listing Method involves listing the multiples of each number until we find the smallest common multiple.

Steps to calculate the LCM of 63, 70, and 77 using the Listing Method:

1. List the multiples:
   • Multiples of 63: 63, 126, 189, 252, 315, 378, 441, 504, 567, 630, 693, 756, 819, 882, 945, 1008, 1071, 1134, 1197, 1260, 1323, 1386, 1449, 1512, 1575, 1638, 1701, 1764, 1827, 1890, 1953, 2016, 2079, 2142, 2205, 2268, 2331, 2394, 2457, 2520, 2583, 2646, 2709, 2772, 2835, 2898, 2961, 3024, 3087, 3150, 3213, 3276, 3339, 3402, 3465, 3528, 3591, 3654, 3717, 3780, 3843, 3906, 3969, 4032, 4095, 4158, 4221, 4284, 4347, 4410, 4473, 4536, 4599, 4662, 4725, 4788, 4851, 4914, 4977, 5040, 5103, 5166, 5229, 5292, 5355, 5418, 5481, 5544, 5607, 5670, 5733, 5796, 5859, 5922, 5985, 6048, 6111, 6174, 6237, 6300, 6363, 6426, 6489, 6552, 6615, 6678, 6741, 6804, 6867, 6930
   • Multiples of 70: 70, 140, 210, 280, 350, 420, 490, 560, 630, 700, 770, 840, 910, 980, 1050, 1120, 1190, 1260, 1330, 1400, 1470, 1540, 1610, 1680, 1750, 1820, 1890, 1960, 2030, 2100, 2170, 2240, 2310, 2380, 2450, 2520, 2590, 2660, 2730, 2800, 2870, 2940, 3010, 3080, 3150, 3220, 3290, 3360, 3430, 3500, 3570, 3640, 3710, 3780, 3850, 3920, 3990, 4060, 4130, 4200, 4270, 4340, 4410, 4480, 4550, 4620, 4690, 4760, 4830, 4900, 4970, 5040, 5110, 5180, 5250, 5320, 5390, 5460, 5530, 5600, 5670, 5740, 5810, 5880, 5950, 6020, 6090, 6160, 6230, 6300, 6370, 6440, 6510, 6580, 6650, 6720, 6790, 6860, 6930, 7000
   • Multiples of 77: 77, 154, 231, 308, 385, 462, 539, 616, 693, 770, 847, 924, 1001, 1078, 1155, 1232, 1309, 1386, 1463, 1540, 1617, 1694, 1771, 1848, 1925, 2002, 2079, 2156, 2233, 2310, 2387, 2464, 2541, 2618, 2695, 2772, 2849, 2926, 3003, 3080, 3157, 3234, 3311, 3388, 3465, 3542, 3619, 3696, 3773, 3850, 3927, 4004, 4081, 4158, 4235, 4312, 4389, 4466, 4543, 4620, 4697, 4774, 4851, 4928, 5005, 5082, 5159, 5236, 5313, 5390, 5467, 5544, 5621, 5698, 5775, 5852, 5929, 6006, 6083, 6160, 6237, 6314, 6391, 6468, 6545, 6622, 6699, 6776, 6853, 6930
2. Find the smallest common multiple.
   • The LCM of 63, 70, and 77 based on listing is 6930.

Hence, the LCM of 63, 70, and 77 is 6930.

Formula for Calculating LCM

There isn't a single formula for all methods, but here's a general approach for two numbers using their Greatest Common Divisor (GCD):

LCM(a, b) = (a * b) / GCD(a, b)

In our case, for three numbers:

LCM(a, b, c) = LCM(LCM(a, b), c)

Conclusion

We have explored different methods to find the LCM of 63, 70, and 77. Whether you use the Prime Factorization Method, Division Method, or Listing Method, each approach leads us to the same result. Practicing these methods will make you more skilled at finding the LCM of any set of numbers.

Frequently Asked Questions

What Is the LCM of 63, 70, and 77?

The LCM of 63, 70, and 77 is 6930.

Can I Use Any Method to Find the LCM?

Yes, you can use the Prime Factorization Method, Division Method, or Listing Method. All will give you the same result.

Why Is Finding the LCM Important?

Finding the LCM is essential in problems that involve adding, subtracting, or comparing fractions. It simplifies these operations by giving a common multiple.