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Determine the prime factorization of the following number:
\[13915\]

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VerifiedVerified
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Hint: To find the prime factors, start by dividing the number by the prime numbers and observe the remainders, if they are equal to 0 or not.

Start by dividing the number by the first prime number, which is $2$. If there is no remainder., it means you can divide evenly, then \[2\] is a factor of the number. Continue dividing by $2$ until you cannot divide evenly anymore. Write down how many \[2's\] you were able to divide the number by evenly. Now try dividing by the next prime factor, which is \[3\] . Ultimately the goal is to get to a quotient of \[1\].

\[13,915{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}6,957.5\] - This has a remainder. Let's try another prime number.
\[13,915{\text{ }} \div {\text{ }}3{\text{ }} = {\text{ }}4,638.3333\] - This has a remainder. Let's try another prime number.
\[13,915{\text{ }} \div {\text{ }}5{\text{ }} = {\text{ }}2,783\] - There is no remainder. Hence, \[5\] is one of the factors.
\[2,783{\text{ }} \div {\text{ }}5{\text{ }} = {\text{ }}556.6\] - There is a remainder. We can't divide by \[5\] evenly anymore. Let's try the next prime number.
\[2,783{\text{ }} \div {\text{ }}7{\text{ }} = {\text{ }}397.5714\] - This has a remainder. \[7\] is not a factor.
\[2,783{\text{ }} \div {\text{ }}11{\text{ }} = {\text{ }}253\] - There is no remainder. Hence, \[11\] is one of the factors.
\[253{\text{ }} \div {\text{ }}11{\text{ }} = {\text{ }}23\] - There is no remainder. Hence, \[11\] is one of the factors.
\[23{\text{ }} \div {\text{ }}11{\text{ }} = {\text{ }}2.0909\] - There is a remainder. We can't divide by \[11\] evenly anymore. Let's try the next prime number
\[23{\text{ }} \div {\text{ }}13{\text{ }} = {\text{ }}1.7692\] - This has a remainder. \[13\] is not a factor.
\[23{\text{ }} \div {\text{ }}17{\text{ }} = {\text{ }}1.3529\] - This has a remainder. \[17\] is not a factor.
\[23{\text{ }} \div {\text{ }}19{\text{ }} = {\text{ }}1.2105\] - This has a remainder. \[19\] is not a factor.
\[23{\text{ }} \div {\text{ }}23{\text{ }} = {\text{ }}1\] - There is no remainder. Hence, \[23\] is one of the factors.

The prime factors of the given number are $5$, $11$, $23$.
As we can see, we can write $13915$ as $5 \times 11 \times 11 \times 23$. It can also be written in exponential form as ${5^1} \times {11^2} \times {23^1}$.

Note: The prime factors of a number are all the prime numbers that, when multiplied together (while also taking in account the number of times they have occurred), equals the original number. You can find the prime factorization of a number by using a factor tree and dividing the number into smaller parts.