What is the prime factorization of 360?

Understand the Problem

The question is asking for the prime factorization of the number 360, which means breaking it down into the prime numbers that multiply together to give 360.

Answer

The prime factorization of 360 is $2^3 \times 3^2 \times 5^1$.

Steps to Solve

Start with the number 360

We will perform the prime factorization by dividing 360 by the smallest prime number. Start with 2, the smallest prime.

Divide by 2

Divide 360 by 2:
$$ 360 \div 2 = 180 $$.
So, one of the prime factors is 2.

Continue with 180

Now, take 180 and divide by the smallest prime number again (still 2):
$$ 180 \div 2 = 90 $$.
We have found another prime factor of 2.

Continue with 90

Next, divide 90 by 2:
$$ 90 \div 2 = 45 $$.
We have another prime factor of 2.

Switch to the next prime number

Now, since 45 is not divisible by 2, we switch to the next smallest prime number, which is 3:
$$ 45 \div 3 = 15 $$.
So, we have found a new prime factor, 3.

Continue with 15

Now divide 15 by 3:
$$ 15 \div 3 = 5 $$.
Again, we have another prime factor of 3.

Finish with 5

Lastly, we can notice that 5 is a prime number, so we stop here.

Combine all prime factors

Now, we write down all the prime factors we gathered:
$$ 360 = 2^3 \times 3^2 \times 5^1 $$

The prime factorization of 360 is $2^3 \times 3^2 \times 5^1$.

More Information

Prime factorization helps in identifying the building blocks of a number. Each prime factor can be multiplied together to obtain the original number, and this is useful in various fields including number theory, cryptography, and simplifying fractions.

Tips

Some students forget to divide completely or re-use prime factors incorrectly. Make sure to divide by the smallest prime repeatedly until you cannot divide anymore before moving to the next prime.
Forgetting to express the factors with their exponents can lead to incomplete answers.

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