Welcome! Today we'll learn about prime factorization by finding the prime factors of four hundred sixty-eight. Prime factorization means breaking down a number into its prime number components. Our method is to divide by the smallest prime numbers until we reach one.
Let's start dividing by the smallest prime number, which is two. Four hundred sixty-eight divided by two equals two hundred thirty-four. Since two hundred thirty-four is still even, we divide by two again. Two hundred thirty-four divided by two equals one hundred seventeen. So far we have two factors of two, giving us two times two times one hundred seventeen.
Now we have one hundred seventeen, which is not divisible by two. Let's try the next prime number, which is three. One hundred seventeen divided by three equals thirty-nine. Since thirty-nine is also divisible by three, we divide again. Thirty-nine divided by three equals thirteen. Now we have two squared times three squared times thirteen.
Finally, we reach thirteen. Thirteen is a prime number because it cannot be divided by any smaller prime numbers like two, three, five, seven, or eleven. Therefore, thirteen is our final prime factor. The complete prime factorization of four hundred sixty-eight is two squared times three squared times thirteen. We can verify this: four times nine times thirteen equals four hundred sixty-eight.
To summarize what we learned: The prime factorization of four hundred sixty-eight is two squared times three squared times thirteen. We used the systematic method of dividing by the smallest prime numbers first. This method works for any composite number, and every number has a unique prime factorization.