IMathAS-libretexts

Factoring and Prime Factorization
Factoring natural numbers is the decomposition of the natural number into a product of other natural numbers, which when multiplied together give the original. For example, the number $\displaystyle {16}$ can be factored into $\displaystyle {2}\cdot{8}$ and $\displaystyle {4}\cdot{4}$ The prime factorization of a natural number is the decomposition of the natural number into a product of a set of prime numbers, which when multiplied together give the original. For example, the number $\displaystyle {12}$ has the prime factorization of $\displaystyle {2}\cdot{2}\cdot{3}$ The number $\displaystyle {24}$ can be factored down to $\displaystyle {2}\cdot{12}$ , $\displaystyle {3}\cdot{8}$ and $\displaystyle {4}\cdot{6}$ The prime factorization of the number $\displaystyle {24}$ is $\displaystyle {2}\cdot{2}\cdot{2}\cdot{3}$ If the variable $\displaystyle {x}$ represents a natural number, then the factorization of $\displaystyle {x}^{{2}}$ is $\displaystyle {x}\cdot{x}$ and the factorization of $\displaystyle {x}^{{3}}$ is $\displaystyle {x}\cdot{x}\cdot{x}$ The prime factorization of $\displaystyle {4}{x}^{{2}}$ is $\displaystyle {2}\cdot{2}\cdot{x}^{{2}}$

Factoring and Prime Factorization

The following is a series of statements about Factoring and The Prime Factorization of Integers. Identify which statement has an error.

Factoring natural numbers is the decomposition of the natural number into a product of other natural numbers, which when multiplied together give the original. For example, the number $\displaystyle {16}$ can be factored into $\displaystyle {2}\cdot{8}$ and $\displaystyle {4}\cdot{4}$
The prime factorization of a natural number is the decomposition of the natural number into a product of a set of prime numbers, which when multiplied together give the original. For example, the number $\displaystyle {12}$ has the prime factorization of $\displaystyle {2}\cdot{2}\cdot{3}$
The number $\displaystyle {24}$ can be factored down to $\displaystyle {2}\cdot{12}$ , $\displaystyle {3}\cdot{8}$ and $\displaystyle {4}\cdot{6}$
The prime factorization of the number $\displaystyle {24}$ is $\displaystyle {2}\cdot{2}\cdot{2}\cdot{3}$
If the variable $\displaystyle {x}$ represents a natural number, then the factorization of $\displaystyle {x}^{{2}}$ is $\displaystyle {x}\cdot{x}$ and the factorization of $\displaystyle {x}^{{3}}$ is $\displaystyle {x}\cdot{x}\cdot{x}$
The prime factorization of $\displaystyle {4}{x}^{{2}}$ is $\displaystyle {2}\cdot{2}\cdot{x}^{{2}}$