Question

What are the prime factors of 12?

Short answer

2, 2, 3

Step-by-step solution

- Understand Prime Factorization

Prime factorization is the process of breaking down a number into the set of prime numbers that, when multiplied together, result in the original number. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves.

- Start with the Smallest Prime Number

Begin by dividing the number 12 by the smallest prime number, which is 2. Since 12 is even, it is divisible by 2: 12 ÷ 2 = 6.

- Factor the Resulting Quotient

Next, factor 6 by dividing it by the smallest prime number: 6 ÷ 2 = 3.

- Check if the Result is Prime

The quotient is now 3, which is a prime number. Therefore, we stop factoring here.

- Write the Prime Factors

Finally, we list all the prime numbers we used in the factorization: 2, 2, and 3.Thus, the prime factors of 12 are 2, 2, and 3.

Key concepts

mathematics

Mathematics is a field of study that explores numbers, quantities, shapes, and patterns. It is a foundational discipline used in various fields such as science, engineering, economics, and everyday life.
When we perform mathematical operations like addition, subtraction, multiplication, and division, we're engaging in fundamental aspects of mathematics.
More advanced concepts include algebra, geometry, calculus, and statistics.
In the context of prime factorization, we use basic arithmetic operations and the unique properties of prime numbers to break down a composite number into its prime factors.
This helps us understand the number's structure and can be useful in various applications, such as encryption in computer science or simplifying fractions in algebra.

prime numbers

Prime numbers are the building blocks of the number system. They are defined as numbers greater than 1 that have no positive divisors other than 1 and themselves.
For example, the prime numbers up to 10 are 2, 3, 5, and 7.
These numbers are essential in prime factorization because they cannot be broken down into smaller factors.
A prime number has exactly two distinct divisors, 1 and the number itself.
Composite numbers, on the other hand, can be divided by numbers other than 1 and themselves.
Understanding prime numbers is crucial for performing prime factorization because we repeatedly divide the composite number by the smallest prime numbers until only primes remain.

factorization

Factorization is the process of breaking down a number into smaller numbers that, when multiplied together, give the original number.
In prime factorization, we specifically break down a composite number into its prime factors.
For instance, in the provided exercise, we factorize 12 by dividing it by the smallest prime numbers until we're left with only primes.
The steps were:
1. Divide 12 by 2 to get 6, since 2 is the smallest prime number.
2. Divide 6 by 2 to get 3, where 3 is another prime number.
Finally, the prime factors of 12 are 2, 2, and 3, since multiplying these primes together (2 * 2 * 3) gives us the original number, 12.
This process simplifies complex numbers and reveals their underlying structure.