Problem 23 - Fundamentals of Mathematics

Question

Find the prime factorization of each whole number. 61

Step-by-Step Solution

Answer

61 is a prime number, so its prime factorization is 61.

Step 1: Understanding Prime Numbers

Prime numbers are numbers that have only two divisors: 1 and themselves. To perform prime factorization, we need to identify whether a number is a prime or composite.

Step 2: Determine if 61 is a Prime Number

To find if 61 is a prime, check divisibility with all prime numbers less than or equal to the square root of 61. Calculate the square root of 61, which is approximately 7.81. Check divisibility with 2, 3, 5, and 7.

Step 3: Check Divisibility with Prime Numbers

1. **Divisibility by 2:** 61 is odd, so it can't be divisible by 2. 2. **Divisibility by 3:** The sum of the digits (6+1=7) is not divisible by 3. 3. **Divisibility by 5:** 61 does not end in 0 or 5. 4. **Divisibility by 7:** Perform division: 61 ÷ 7 ≈ 8.71, not an integer.

Step 4: Conclusion About 61

Since 61 is not divisible by any prime number less than or equal to its square root, 61 is a prime number.

Key Concepts

Prime Numbers

Prime numbers are special due to their unique property of having only two divisors: 1 and themselves. These numbers cannot be broken down or divided by any other numbers without leaving a remainder. For instance, 2, 3, 5, 7, and 11 are all prime numbers. Each of these has exactly two divisors.
Understanding prime numbers is a key step in learning prime factorization, which involves expressing a number as a product of prime numbers. To determine if a number is prime, first evaluate its factors. When checking if a number is prime, consider all numbers up to its square root because if it's not divisible by any of these, it has no other divisors. For example, to see if 61 is a prime number, check divisibility with prime numbers up to its square root, approximately 7.8.
Checking these smaller numbers simplifies the process since confirming divisibility beyond this point is unnecessary.

Divisibility Rules

Divisibility rules are handy shortcuts to quickly determine if a number is divisible by another. These rules save time when testing large numbers for prime factorization. Some commonly used divisibility rules include:

• Divisibility by 2: A number is divisible by 2 if it is even, meaning its last digit is 0, 2, 4, 6, or 8.
• Divisibility by 3: Check if the sum of the digits of the number is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 7: To check divisibility by 7, divide the number by 7 and see if the quotient is an integer.

Using these rules helps quickly narrow down whether a number is prime or composite, particularly when performing steps such as identifying the factors of 61. Since 61 isn't divisible by numbers like 2, 3, 5, or even 7, it confirms its status as a prime number.

Composite Numbers

Unlike prime numbers, composite numbers have more than two divisors. This means they can be divided exactly by numbers other than 1 and themselves. For instance, the number 4 is composite because it can be divided by 1, 2, and 4.
Identifying a number as composite is essential in prime factorization because composites can be broken down into a product of prime numbers. A composite number results from multiplying prime numbers together. For example, the number 12 can be expressed as 2 × 2 × 3 — a series of prime factors.
Understanding the distinction between prime and composite numbers is key in mathematics, especially when attempting to solve problems related to factorization. Recognizing that 61 isn't composite reaffirms its status as a prime number, showcasing the efficiency of these fundamental concepts in confirming the nature of numbers.