Key takeaway: The prime factorization of a number is the way of writing it as a product of prime numbers, which is unique apart from order, such as 756 = 2 × 2 × 3 × 3 × 3 × 7.
Grade 66.NS.B.4Numbers and Operations
Definition of Prime Factorization
Prime factorization breaks a composite number into prime building blocks. A prime has exactly two factors, 1 and itself, such as 2, 3, 5, 7 and 11.
A factor tree is a handy way to find it: split the number into any two factors, then keep splitting each composite factor until only primes are left. Every whole number greater than 1 has one such product, no matter which splits you choose first. Repeated primes are shortened with exponents.
Formula and notation
Primes are multiplied and repeated ones get exponents:
\[756 = 2 \times 2 \times 3 \times 3 \times 3 \times 7 = 2^2 \times 3^3 \times 7\]
The exponent counts how many times that prime appears in the product.
Key facts
- Different factor trees for the same number always end with the same primes.
- The number 1 is not prime, so it never appears in the factorization.
- Prime factorization gives the GCF and LCM quickly by comparing exponents.
- A number is a perfect square when every exponent in its factorization is even.
Prime Factorization example
Write the prime factorization of 756 and use it to decide whether 756 is a perfect square.
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Split with the smallest primes: \(756 = 2 \times 378\), \(378 = 2 \times 189\).
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Continue: \(189 = 3 \times 63\), \(63 = 3 \times 21\), \(21 = 3 \times 7\).
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Collect: \(756 = 2^2 \times 3^3 \times 7\). Check: \(4 \times 27 \times 7 = 756\).
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The exponents 3 and 1 are odd, so 756 is not a perfect square.
The factorization is \(2^2 \times 3^3 \times 7\), and 756 is not a perfect square.
Common mistake
Watch out
Ivy writes \(756 = 2^2 \times 27 \times 7\) and says she is done.
Why it is wrong
The product is correct, but 27 is not prime, since \(27 = 3 \times 3 \times 3\). A true factorization uses only primes, so 27 must be split into \(3^3\). Test each factor: if it has a divisor other than 1 and itself, keep factoring.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations