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There are sometimes a number of methods to speak about the identical concept. Thus far, we’ve seen that if [latex]m[/latex] is a a number of of [latex]n[/latex], we are able to say that [latex]m[/latex] is divisible by [latex]n[/latex]. We all know that [latex]72[/latex] is the product of [latex]8[/latex] and [latex]9[/latex], so we are able to say [latex]72[/latex] is a a number of of [latex]8[/latex] and [latex]72[/latex] is a a number of of [latex]9[/latex]. We will additionally say [latex]72[/latex] is divisible by [latex]8[/latex] and by [latex]9[/latex]. One other strategy to speak about that is to say that [latex]8[/latex] and [latex]9[/latex] are elements of [latex]72[/latex]. Once we write [latex]72=8cdot 9[/latex] we are able to say that we now have factored [latex]72[/latex].

In algebra, it may be helpful to find out all the elements of a quantity. That is known as factoring a quantity, and it might assist us resolve many sorts of issues.

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For instance, suppose a choreographer is planning a dance for a ballet recital. There are [latex]24[/latex] dancers, and for a sure scene, the choreographer needs to rearrange the dancers in teams of equal sizes on stage.

In what number of methods can the dancers be put into teams of equal measurement? Answering this query is similar as figuring out the elements of [latex]24[/latex]. The desk beneath summarizes the totally different ways in which the choreographer can organize the dancers.

**Variety of Teams** **Dancers per Group** **Complete Dancers** [latex]1[/latex] [latex]24[/latex] [latex]1cdot 24=24[/latex] [latex]2[/latex] [latex]12[/latex] [latex]2cdot 12=24[/latex] [latex]3[/latex] [latex]8[/latex] [latex]3cdot 8=24[/latex] [latex]4[/latex] [latex]6[/latex] [latex]4cdot 6=24[/latex] [latex]6[/latex] [latex]4[/latex] [latex]6cdot 4=24[/latex] [latex]8[/latex] [latex]3[/latex] [latex]8cdot 3=24[/latex] [latex]12[/latex] [latex]2[/latex] [latex]12cdot 2=24[/latex] [latex]24[/latex] [latex]1[/latex] [latex]24cdot 1=24[/latex]

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What patterns do you see within the desk above? Did you discover that the variety of teams occasions the variety of dancers per group is all the time [latex]24?[/latex] This is sensible, since there are all the time [latex]24[/latex] dancers.

It’s possible you’ll discover one other sample should you look rigorously on the first two columns. These two columns include the very same set of numbers—however in reverse order. They’re mirrors of each other, and in reality, each columns listing all the elements of [latex]24[/latex], that are:

[latex]1,2,3,4,6,8,12,24[/latex]

We will discover all of the elements of any counting quantity by systematically dividing the quantity by every counting quantity, beginning with [latex]1[/latex]. If the quotient can also be a counting quantity, then the divisor and the quotient are elements of the quantity. We will cease when the quotient turns into smaller than the divisor.

Within the following video we present easy methods to discover all of the elements of [latex]30[/latex].

### Establish Prime and Composite Numbers

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Some numbers, like [latex]72[/latex], have many elements. Different numbers, comparable to [latex]7[/latex], have solely two elements: [latex]1[/latex] and the quantity. A quantity with solely two elements known as a main quantity. A quantity with greater than two elements known as a composite quantity. The quantity [latex]1[/latex] is neither prime nor composite. It has just one issue, itself.

The desk beneath lists the counting numbers from [latex]2[/latex] by [latex]20[/latex] together with their elements. The highlighted numbers are prime, since every has solely two elements.

Components of the counting numbers from [latex]2[/latex] by [latex]20[/latex], with prime numbers highlighted

The prime numbers lower than [latex]20[/latex] are [latex]2,3,5,7,11,13,17,textual content{and }19[/latex]. There are various bigger prime numbers too. As a way to decide whether or not a quantity is prime or composite, we have to see if the quantity has any elements apart from [latex]1[/latex] and itself. To do that, we are able to check every of the smaller prime numbers with a purpose to see if it’s a issue of the quantity. If not one of the prime numbers are elements, then that quantity can also be prime.

Within the following video we present extra examples of easy methods to decide whether or not a quantity is prime or composite.

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