Find the prime factorization of 210.

Understand the Problem

The question is asking to find the prime factors of the number 210. This involves determining which prime numbers multiply together to give the original number. The process will involve dividing 210 by the smallest prime numbers until we have completely factored it into primes.

Answer

The prime factors of 210 are $ 2, 3, 5, $ and $ 7 $.

Steps to Solve

Start with the number 210

We begin with the number 210 and will try dividing it by the smallest prime number, which is 2.

Divide by the smallest prime

We check if 210 is even. Since it is, we divide it by 2:

$$ 210 \div 2 = 105 $$

Now we have factored out 2.

Continue with the next smallest prime

Next, we take 105. We need to test divisibility by the next prime, which is 3.

To check if 105 is divisible by 3, we sum the digits (1 + 0 + 5 = 6), and since 6 is divisible by 3, we proceed to divide:

$$ 105 \div 3 = 35 $$

Now we have factored out 3.

Proceed with the next prime

Now we factor 35. The next prime is 5. Since 35 ends in 5, it is divisible by 5:

$$ 35 \div 5 = 7 $$

We have factored out 5.

Identify the last prime factor

Now we have 7, which is also a prime number. We cannot divide further, and thus we stop here.

Compile the prime factors

Now, we can write the final list of prime factors of 210. The complete factorization is:

$$ 210 = 2 \times 3 \times 5 \times 7 $$

The prime factors of 210 are: ( 2, 3, 5, ) and ( 7 ).

More Information

The number 210 is composed of multiple prime factors, which are the building blocks of all whole numbers. Prime factorization is a fundamental concept in number theory and has applications in areas such as cryptography.

Tips

Forgetting to check divisibility by smaller primes first.
Miscalculating the division when breaking down a number, which can lead to incorrect factors.
Not recognizing when to stop factoring—once you reach a prime number, you don't need to factor further.

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