Hi, and welcome come this video clip covering the least common multiple and the greatest usual factor!

As girlfriend know, there are times when we need to algebraically “adjust” exactly how a number or an equation appears in bespeak to continue with our mathematics work. We deserve to use the greatest typical factor and the least usual multiple to perform this. The greatest typical factor (GCF) is the biggest number that is a aspect of 2 or an ext numbers, and also the least usual multiple (LCM) is the smallest number that is a lot of of 2 or an ext numbers.

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To see just how these principles are useful, let’s look at adding fractions. Prior to we can include fractions, we need to make certain the denominators room the very same by creating an indistinguishable fraction:

$$\frac23+\frac16 \rightarrow \frac23 \times \frac22$$$$+\frac16 \rightarrow \frac46 +\frac16=\frac56$$

In this example, the least common multiple the 3 and 6 must be determined. In various other words, “What is the smallest number the both 3 and also 6 deserve to divide into evenly?” with a small thought, we realize the 6 is the least usual multiple, since 6 divided by 3 is 2 and also 6 split by 6 is 1. The portion $$\frac23$$ is then adjusted to the equivalent portion $$\frac46$$ by multiply both the numerator and also denominator by 2. Currently the 2 fractions with common denominators deserve to be included for a last value that $$\frac56$$.

In the context of adding or subtracting fractions, the least common multiple is described as the least usual denominator.

In general, you need to identify a number bigger than or equal to two or much more numbers to find their least usual multiple.

It is vital to note that over there is an ext than one method to determine the least usual multiple. One means is to simply list every the multiples of the values in question and select the smallest common value, as checked out here:

Least usual multiple that 8, 4, 6
$$8\rightarrow 8,16,24,32,40,48$$ $$4\rightarrow 4,8,12,16,20,24,28,32$$ $$6\rightarrow 6,12,18,24,30,36$$

This illustrates the the least typical multiple of 8, 4, and 6 is 24 due to the fact that it is the the smallest number that 8, 4, and also 6 can all divide into evenly.

Another common method involves the prime factorization of each value. Remember, a element number is only divisible by 1 and itself.

Once the prime components are determined, perform the shared factors once, and also then main point them through the other remaining element factors. The an outcome is the least common multiple:

$$30=2\times 2\times 3\times 3$$ $$90=2\times 3\times 3\times 5$$
$$\textLCM=2\times 3\times 3\times 2\times 5$$

The least typical multiple can additionally be discovered by common (or repeated) division. This an approach is sometimes taken into consideration faster and much more efficient than listing multiples and also finding element factors. Right here is an instance of recognize the least common multiple that 3, 6, and also 9 utilizing this method:

Divide the number by the determinants of any type of of the 3 numbers. 6 has a variable of 2, for this reason let’s usage 2. Nine and also 3 cannot be split by 2, for this reason we’ll simply rewrite 9 and 3 here. Repeat this procedure until every one of the numbers are reduced to 1. Then, multiply all of the factors together to obtain the least usual multiple.

 2 3 6 9 3 3 3 9 3 1 1 3 1 1 1
LCM $$=2\times 3\times 3=18$$

Now that techniques for finding least typical multiples have been introduced, we’ll require to adjust our mindset to detect the greatest usual factor of 2 or an ext numbers. We will be identify a value smaller than or same to the numbers gift considered. In various other words, asking yourself, “What is the largest value that divides both of these numbers?” expertise this ide is vital for dividing and also factoring polynomials.

Prime administer can also be supplied to determine the greatest common factor. However, fairly than multiplying every the prime factors like us did because that the least typical multiple, we will certainly multiply only the prime factors that the number share. The resulting product is the greatest usual factor.

## Review

stop wrap up with a pair of true or false review questions:

1. The least common multiple that 45 and also 60 is 15.Show Answer

The greatest usual factor of 45 and 60 is 15, however the least common multiple is 180.

2. The least typical multiple is a number higher than or equal to the numbers being considered. Show Answer

The price is true.

The least typical multiple is higher than or equal to the numbers gift considered, when the greatest common factor is equal to or less than the numbers being considered.

Thanks because that watching, and happy studying!

A

There are a variety of approaches for finding the LCM and also GCF. The 2 most usual strategies involve do a list, or using the prime factorization.

for example, the LCM the 5 and 6 can be uncovered by just listing the multiples that $$5$$ and $$6$$, and also then identify the shortest multiple mutual by both numbers.$$5, 10, 15, 20, 25, \mathbf30, 35…$$ $$6, 12, 18, 24, \mathbf30, 36…$$ $$\mathbf30$$ is the LCM.

Similarly, the GCF can be discovered by listing the determinants of every number, and also then identify the greatest variable that is shared. Because that example, the GCF the $$40$$ and also $$32$$ can be discovered by listing the factors of every number.

$$40$$: $$1, 2, 4, 5, \mathbf8, 10, 20, 40$$ $$32$$: $$1, 2, 4, \mathbf8, 16, 32$$ $$\mathbf8$$ is the GCF.

For larger numbers, it will not it is in realistic to make a list of components or multiples to recognize the GCF or LCM. For huge numbers, it is most effective to usage the element factorization technique.

for example, as soon as finding the LCM, begin by recognize the element factorization of every number (this deserve to be done by creating a variable tree). The prime factorization the $$20$$ is $$2\times2\times5$$, and the prime factorization of $$32$$ is $$2\times2\times2\times2\times2$$. Circle the factors that space in common and also only counting these once.

now multiply every one of the determinants (remember no to double-count those circled $$2$$s). This i do not care $$2\times2\times5\times2\times2\times2$$, which equates to $$160$$. The LCM of $$20$$ and $$32$$ is $$160$$.

once finding the GCF, start by listing the prime factorization of every number (this deserve to be done by developing a element tree). For example, the element factorization the $$45$$ is $$5\times3\times3$$, and the element factorization of $$120$$ is $$5\times3\times2\times2\times2$$. Now simply multiply every one of the determinants that are mutual by both numbers. In this case, we would multiply $$5\times3$$ which equates to $$15$$. The GCF the $$45$$ and $$120$$ is $$15$$.

The element factorization strategy can seem like a fairly an extensive process, but when functioning with large numbers it is guaranteed to it is in a time-saver.

A

There are two main strategies for finding the GCF: Listing the factors, or utilizing the element factorization.

The an initial strategy entails simply listing the components of every number, and also then in search of the greatest variable that is common by both numbers. Because that example, if we are searching for the GCF the $$36$$ and also $$45$$, we can list the determinants of both numbers and also identify the biggest number in common. $$36$$: $$1,2,3,4,6,\mathbf9,12,18,36$$ $$45$$: $$1,3,5,\mathbf9,15,45$$ The GCF of $$36$$ and $$45$$ is $$\mathbf9$$.

Listing the factors of every number and then identifying the largest element in typical works well for little numbers. However, as soon as finding the GCF the very large numbers that is an ext efficient to use the prime factorization approach.

for example, as soon as finding the GCF the $$180$$ and also $$162$$, we begin by listing the prime factorization of every number (this can be excellent by producing a factor tree). The element factorization of $$180$$ is $$2\times2\times3\times3\times5$$, and the prime factorization that $$162$$ is $$2\times3\times3\times3\times3$$. Now look because that the components that are common by both numbers. In this case, both number share one $$2$$, and two $$3$$s, or $$2\times3\times3$$. The result of $$2\times3\times3$$ is $$18$$, i beg your pardon is the GCF! This strategy is often an ext efficient once finding the GCF that really huge numbers.

A

There are a variety of approaches for detect the lowest usual multiple. Two common approaches are listing the multiples, and also using the prime factorization. Listing the multiples is simply as that sounds, merely list the multiples of each number, and also then look because that the shortest multiple shared by both numbers. For example, when finding the lowest typical multiple of $$3$$ and $$4$$, list the multiples: $$3$$: $$3,6,9,\mathbf12,15,18…$$ $$4$$: $$4,8,\mathbf12,16,20…$$ $$\mathbf12$$ is the shortest multiple mutual by $$3$$ and also $$4$$.

Listing the multiples is a great strategy when the numbers are fairly small. Once numbers are large, such together $$38$$ and $$42$$, we must use the prime factorization approach. Begin by listing the prime factorization of every number (this deserve to be done making use of a factor tree). $$38$$: $$2\times19$$ $$42$$: $$2\times3\times7$$ now circle the shared determinants (only count these once).

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currently multiply every one of the determinants (remember to just count the $$2$$s once). This i do not care $$2\times19\times3\times7$$, which amounts to $$798$$. The LCM of $$38$$ and $$42$$ is $$798$$.