Write prime factorizations

Prime factorization represents a whole number greater than 1 as a product of prime numbers, including repeated factors, with 1 correctly excluded from the primes; exponent notation records repeated prime factors compactly. The factorization is found by decomposing composite factors and is unique apart from the order of the factors, supporting divisibility, greatest common factor, least common multiple, and fraction work; advanced proofs and factorization of algebraic expressions are not included.

Detailed Explanation: Write 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 factors: $1$ and itself. Remember, $1$ is not prime.

To find a prime factorization:

Start with the number.
Break it into factors.
Keep breaking any composite factors.
Stop when every factor is prime.
Use exponents when the same prime factor is repeated.

Example: Find the prime factorization of $60$

First, break $60$ into two factors:

60=6\cdot 10

Both $6$ and $10$ are composite, so break them down:

6=2\cdot 3

10=2\cdot 5

Substitute these into the original factorization:

60=(2\cdot 3)(2\cdot 5)

Rearrange the factors to group the repeated prime:

60=2\cdot 2\cdot 3\cdot 5

The prime factor $2$ appears twice, so write it with an exponent:

\boxed{60=2^2\cdot 3\cdot 5}

Check that all the factors are prime: $2$ , $3$ , and $5$ are prime. Also,

2^2\cdot 3\cdot 5=4\cdot 3\cdot 5=60

Therefore, the prime factorization of $60$ is $\boxed{2^2\cdot 3\cdot 5}$ .

Learn by doing: Write prime factorizations

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Prime Factorization - Factor Tree with 3 Factors - Full

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