Write the prime factorization of 165. Use exponents when appropriate and order the factors from least to greatest (for example, [tex]$2^2 \cdot 3 \cdot 5$[/tex]).
Answer (2)
We need to find the prime factorization of 165.
Divide 165 by the smallest prime number, 3, to get 55: 165 = 3 ⋅ 55
Divide 55 by the next smallest prime number, 5, to get 11: 55 = 5 ⋅ 11
Since 11 is a prime number, the prime factorization of 165 is: 3 ⋅ 5 ⋅ 11
Explanation
Understanding the Problem We are asked to find the prime factorization of 165, expressing it with exponents when a prime factor appears more than once, and ordering the factors from least to greatest.
Testing Divisibility by 2 We will start by dividing 165 by the smallest prime number, 2. Since 165 is odd, it is not divisible by 2.
Finding the First Prime Factor Next, we try dividing 165 by the next smallest prime number, 3. 165 ÷ 3 = 55 So, 3 is a factor of 165.
Finding Factors of 55 Now we need to find the prime factors of 55. We start by checking if 55 is divisible by 3. It is not.
Finding the Second Prime Factor We try dividing 55 by the next prime number, 5. 55 ÷ 5 = 11 So, 5 is a factor of 55.
Identifying the Last Prime Factor Now we have 11, which is itself a prime number.
Writing the Prime Factorization Therefore, the prime factors of 165 are 3, 5, and 11. We write the prime factorization as: 3 ⋅ 5 ⋅ 11
Final Answer The prime factorization of 165, ordered from least to greatest, is 3 ⋅ 5 ⋅ 11 .
Examples
Prime factorization is a fundamental concept in number theory with many practical applications. For example, when creating secure encryption algorithms, large numbers are often used, and the difficulty of factoring these numbers into their prime components is what makes the encryption strong. In real life, this ensures secure online transactions and protects sensitive data.
The prime factorization of 165 is expressed as 3 ⋅ 5 ⋅ 11 . This is determined through systematic division by prime numbers until reaching only prime factors. In this case, 3, 5, and 11 are the prime factors of 165.
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