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A multiple is a number that may be divided by another number without leaving a trail. As an illustration:
5 times 4 = 20
Or, 5 × 4 = 20
And, 20 ÷ 5 = 4
Least common multiple
The “smallest non-zero common number” between two numbers is the “least common multiple.”
LCM example
There are several methods for calculating the LCM of numbers. Finding the least common multiple of two numbers can be done in one of three ways. Following are some instances of each method.
The following are some of the methods for determining the least common multiple of two or more numbers:
- Using prime factorization
- Using repeated division
- Using multiples
LCM using prime factorization
A factorization tree is constructed for each given number by listing the multiples of that number in this technique. The least prime factors for that number are found on the tree’s last branch.
The factorization trees for 36 and 48, for example, look like this:
LCM = 2 × 2 × 3 × 3 × 2 × 2
LCM = 144
2. LCM using repeated division
The specified integers are divided by the common divisors until no further division by the common number is possible using this method. To find the LCM, multiply the divisors and the remainders together.
LCM = 2 × 2 × 3 × 4 × 3
LCM = 144
3. LCM using multiples
List the multiples of the numbers in the table, as shown, to find the LCM using multiples. For the given numbers, the least common multiple is the same as the first common multiple.
For 36 and 48, the number 144 is the LCM.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ||||
| 36 | 72 | 108 | 144 | 180 | 216 | 252 | 288 | 324 | 360 | 396 | 432 | ||||
| 48 | 96 | 144 | 192 | 240 | 288 | 336 | 384 | 432 | 480 | 528 | 576 | ||||
Application
The dimension of using the LCM of two integers begins with simple fractional math operations like addition and subtraction. The LCM value can be used to optimize the amounts of two things in math problems where they are paired against each other. The LCM of numbers also aids in the construction of encrypted messages in computer science.
Question: Find the LCM of 2,3,7,21.
Solution: The greatest number should be chosen. Number 21 is the most. Examine the remaining numbers to see if 21 is divisible by them. The numbers 3 and 7 are divisible by 21, but not by 2. As a result, divide 21 by 2. 42 is the outcome. Now see if 42 can be divided by 2, 3, or 7. 42 can be divided in two ways. As a result, 42 is the LCM.
LCM Formulas
Numbers, their LCM, and their HCF are all combined in LCM formulas (Highest Common Factor). The least common multiple of two integers and the LCM of two fractions are calculated using these formulas. For integers and fractions, the LCM formulas are presented below.
LCM Formula for Integers
The formula for the least common multiple of two numbers is:
LCM (a,b) = (a x b)/HCF(a,b)
LCM Formula for Fractions
If a/b and c/d is the least common multiple of the two fractions, then the formula is:
LCM (a/b, c/d) = (LCM of Numerators)/(HCF of Denominators) = LCM (a,c)/HCF (b,d)
The least common multiple (LCM) and the highest common factor (HCF) between two numbers are the least common multiple and the highest common factor, respectively.
The major goal of demonstrating the difference between these two numbers is to demonstrate the distinction between a factor and a multiple.
An integer that occurs in the timetable is a multiple of a whole number.
Consider the multiples of three, for example:
3, 6, 9, 12, 15, 18, and so on.
The factor of an integer, on the other hand, is the number that divides the integer without leaving any remainder.
Consider the 36 factors, for example:
1, 2, 3, 4, 6, 9, 12, 18, and 36.
To get the LCM and HCF of two different numbers, we must first find the highest common factor of 15 and 18, which is 3.
15 and 18 have an LCM of 90.
L.C.M and H.C.F = 90 X 3 = 270
15 x 18 = 270 is the result of multiplying these two numbers.
As a result, the product of L.C.M and H.C.F is the same as the sum of these two quantities.
Conclusion
The term “Least Common Multiple” is abbreviated as LCM. The smallest number that may be divided by both numbers is called the least common multiple of two numbers. It can be done with two or more integers or fractions.
To find the LCM of two numbers, there are several techniques. The product of the highest powers of the common prime factors is the LCM of those numbers, which is one of the easiest ways to find it. In mathematics, the least common multiple (or LCM) is referred to as the lowest common multiple. The smallest number among all possible multiples of two or more numbers is the least frequent multiple.
