Problem 16 - Elementary Algebra

Question

For the following problems, find the prime factorization of each whole number. Use exponents on repeated factors. 26

Step-by-Step Solution

Answer

Answer: The prime factorization of 26 is 2 × 13.

Step 1: Start with the smallest prime number

Since 26 is an even number, the smallest prime number (2) can divide 26. Divide 26 by 2 to see how many times it can be divided: 26 ÷ 2 = 13

Step 2: Check if the result is a prime number

Now we need to check if 13 is a prime number. A prime number can only be divided by 1 and itself. In this case, 13 is only divisible by 1 and 13, so it is indeed a prime number.

Step 3: Write the prime factorization

Since we have found the prime factors of 26 to be 2 and 13, we can write the prime factorization as: 26 = 2 × 13

Step 4: Check for repeated factors

There are no repeated factors in this case, as the prime factors are 2 and 13 (which are distinct). Thus, the prime factorization of 26 is 2 × 13.

Key Concepts

Prime Numbers

Understanding prime numbers is fundamental to grasping the concept of prime factorization. A prime number is a whole number greater than 1 that has no positive divisors other than 1 and itself. This exclusive club includes numbers like 2, 3, 5, 7, 11, and 13, to name a few.

In our exercise where we find the prime factorization of 26, we identify that 26 is divisible by 2, implying that 2 is one of its prime factors. Importantly, it's also the smallest prime number. Following our factorization process, we then discover that 13 is the next factor, which cannot be divided any further except by 1 or 13, making it a prime number. Recognizing prime numbers aids significantly in factorizing composite numbers into their prime components.

Exponents

Moving onto exponents, they are a shorthand way to show how many times a number, known as the base, is multiplied by itself. For instance, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \). Exponents play a big role in simplifying the notation of prime factorization especially when dealing with repeated factors.

Although our exercise example, 26, did not have repeated factors, let's consider a number like 8. The prime factorization of 8 is 2 multiplied by itself three times, which is elegantly expressed as \( 2^3 \). If we had a scenario where our number had repeated prime factors, using exponents would make our expression much more concise and easier to work with.

Divisibility Rules

Lastly, divisibility rules are useful shortcuts that help us determine whether a number is divisible by another without performing the division. For example, if a number ends in 0, 2, 4, 6, or 8, it's divisible by 2. If the sum of a number's digits is divisible by 3, then the number itself is divisible by 3.

These rules streamline the prime factorization process. In the case of 26, we applied the rule for divisibility by 2: since 26 is even, we knew it was divisible by 2. As we progress to larger numbers, these rules become incredibly valuable, enabling us to quickly bypass possible non-factors and hone in on the actual prime factors of a number.