Determine prime factorizations

Prime factorization expresses a whole number greater than 1 as a product of prime numbers, with repeated factors represented compactly using exponents; 1 is neither prime nor composite, and each number has one unique prime factorization apart from the order of its factors. This understanding supports finding common factors and multiples, simplifying fractions, and reasoning with exponents, but does not extend to factoring algebraic expressions or working in generalized number systems.

Detailed Explanation: Determine prime factorizations

A prime factorization writes a whole number greater than $1$ as a product of only prime numbers. A prime number has exactly two positive factors: $1$ and itself. Examples include $2, 3, 5,$ and $7$ .

To find a prime factorization:

Divide by the smallest prime factor possible.
Keep dividing the quotient by prime numbers.
Stop when all the factors are prime.
Use exponents to write repeated prime factors compactly.

Example: Find the prime factorization of $84$

Start with the smallest prime, $2$ :

84=2\cdot 42

The quotient, $42$ , is still even, so divide by $2$ again:

42=2\cdot 21

Now factor $21$ . It is divisible by $3$ :

21=3\cdot 7

Both $3$ and $7$ are prime. Therefore,

84=2\cdot 2\cdot 3\cdot 7

Because $2$ appears twice, write it using an exponent:

\boxed{84=2^2\cdot 3\cdot 7}

This is the prime factorization of $84$ . The order of the prime factors may change, but the prime factorization itself is unique. Remember that $1$ is neither prime nor composite, so it is not included as a prime factor.

Learn by doing: Determine prime factorizations

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Prime Factorization as Exponents - 5 Factors

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