Write the prime factorization of 147. Use exponents when appropriate and order the factors from least to greatest (for example, [tex]$2^2 \cdot 3 \cdot 5$[/tex]).
Answer (2)
Divide 147 by the smallest prime number, 2. Since 147 is odd, it is not divisible by 2. Try the next smallest prime number, 3. 147 ÷ 3 = 49 .
Factor 49. 49 is not divisible by 5. Divide 49 by 7: 49 ÷ 7 = 7 .
Factor 7. 7 ÷ 7 = 1 .
The prime factors of 147 are 3, 7, and 7. The prime factorization of 147 is 3 ⋅ 7 2 .
Explanation
Problem Analysis We are asked to find the prime factorization of 147, expressing it with exponents and ordering the factors from least to greatest.
Finding the First Prime Factor To find the prime factorization, we start by dividing 147 by the smallest prime number, 2. Since 147 is odd, it is not divisible by 2. The next smallest prime number is 3. We divide 147 by 3: 147 ÷ 3 = 49 . So, 3 is a factor of 147.
Finding the Next Prime Factor Now we need to factor 49. We try dividing by the next prime number, 5. Since 49 does not end in 0 or 5, it is not divisible by 5. The next prime number is 7. We divide 49 by 7: 49 ÷ 7 = 7 . So, 7 is a factor of 49.
Finding the Last Prime Factor Now we need to factor 7. The only prime factor of 7 is 7 itself: 7 ÷ 7 = 1 .
Writing the Prime Factorization So, the prime factors of 147 are 3, 7, and 7. We can write this as 3 ⋅ 7 ⋅ 7 , which can be expressed with exponents as 3 ⋅ 7 2 .
Final Answer Therefore, the prime factorization of 147, with exponents and ordered from least to greatest, is 3 ⋅ 7 2 .
Examples
Prime factorization is a fundamental concept in number theory and has 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 is used to protect sensitive data like credit card numbers and personal information when you make online purchases.
The prime factorization of 147 is expressed as 3 ⋅ 7 2 , which means that 147 can be broken down into the prime factors 3 and 7 (the latter occurring twice). We found this by dividing 147 first by 3, then factoring 49 into 7 and 7. Therefore, the answer is 3 ⋅ 7 2 .
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