Understanding Prime Factors and Factorization

Finding the Prime factors of a number means breaking it down into a product of only prime numbers. Prime factors are essential building blocks in number theory.

A prime number is a whole number greater than 1 that has only two positive divisors: 1 and itself (examples: 2, 3, 5, 7, 11...). A composite number is a whole number greater than 1 that has more than two positive divisors (examples: 4, 6, 8, 9, 10...).

Finding the Prime Factorization of 420

To find the Prime factors of 420, we use the process of prime factorization. We start by dividing 420 by the smallest prime number possible and continue until we are left with only prime numbers.

Here are the steps for the factorization of 420:

• Divide 420 by 2: $420 \div 2 = 210$
• Divide 210 by 2: $210 \div 2 = 105$
• 105 is not divisible by 2. Divide 105 by the next prime number, 3: $105 \div 3 = 35$
• 35 is not divisible by 3. Divide 35 by the next prime number, 5: $35 \div 5 = 7$
• 7 is a prime number. Divide 7 by 7: $7 \div 7 = 1$

When we reach 1, the process is complete. The numbers we divided by are the prime factors.

So, the prime factorization of 420 is $2 \times 2 \times 3 \times 5 \times 7$. These are the Prime factors of 420.

Evaluating Options for the Correct Prime Factors

Now, let's look at the given options and see which one matches our result. The correct option must list only prime numbers that multiply together to give 420.

• Option 1: $2 \times 2 \times 7 \times 15$. This list includes 15. 15 is a composite number ($15 = 3 \times 5$). So, this is incorrect.
• Option 2: $2 \times 2 \times 5 \times 21$. This list includes 21. 21 is a composite number ($21 = 3 \times 7$). So, this is incorrect.
• Option 3: $2 \times 6 \times 5 \times 7$. This list includes 6. 6 is a composite number ($6 = 2 \times 3$). So, this is incorrect.
• Option 4: $2 \times 2 \times 3 \times 5 \times 7$. All numbers in this list (2, 3, 5, 7) are prime numbers. Let's check their product: $2 \times 2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7 = 12 \times 5 \times 7 = 60 \times 7 = 420$. This matches the original number 420, and all factors are prime. This represents the correct prime factorization of 420.

Therefore, the correct list of Prime factors for 420 is $2 \times 2 \times 3 \times 5 \times 7$.