Problem 77 - Prealgebra

Question

Determine whether each number is prime or composite. $$79$$

Step-by-Step Solution

Answer

79 is a prime number.

Step 1: Understanding Prime and Composite Numbers

A prime number is a natural number greater than 1 that can only be divided by 1 and itself. A composite number has factors other than 1 and itself. Our task is to classify the number 79 as either prime or composite.

Step 2: Checking Divisibility by Smaller Primes

To determine if 79 is a prime, we need to check if it is divisible by any smaller prime numbers. The prime numbers to check are 2, 3, 5, and 7 since 79 is less than 100. We'll check each for divisibility.

Step 3: Divisibility Test for 2

Since 79 is an odd number, it is not divisible by 2. We can skip this divisor.

Step 4: Divisibility Test for 3

To test divisibility by 3, sum the digits of 79. The sum is 7 + 9 = 16. Since 16 is not divisible by 3, neither is 79.

Step 5: Divisibility Test for 5

A number is divisible by 5 if its last digit is 0 or 5. 79 ends in 9, so it is not divisible by 5.

Step 6: Divisibility Test for 7

For divisibility by 7, divide 79 by 7. Calculating, we have 79 ÷ 7 ≈ 11.2857, which is not an integer, so 79 is not divisible by 7.

Step 7: Conclusion on Divisibility

Since 79 is not divisible by any of the prime numbers less than its square root, it has no divisors other than 1 and itself. This means 79 is a prime number.

Key Concepts

Divisibility Rules

Divisibility rules are handy shortcuts that help determine if one number divides another without performing full division calculations. These rules simplify checks and save time.
For instance:

• Divisibility by 2: A number is divisible by 2 if it's even. That means it ends in 0, 2, 4, 6, or 8.
• Divisibility by 3: Add the digits of the number. If the sum is divisible by 3, so is the number itself.
• Divisibility by 5: A number will be divisible by 5 if it ends in 0 or 5.
• Divisibility by 7: It's a bit trickier—there are several ways, but checking whether dividing by 7 results in a whole number is straightforward for smaller numbers.

For example, in our exercise with 79, we used these rules to check whether any small prime numbers divide 79 cleanly, helping us identify its nature as a prime number.

Identifying Prime Numbers

Identifying a prime number means determining whether a number can only be divided by 1 and itself. Prime numbers are the building blocks of the entire number system since every number is either a prime or can be reduced to primes (in factorized form). Finding prime numbers requires checking divisibility by all prime numbers up to the number’s square root.
For 79:

• Check with smaller primes: 2, 3, 5, 7 since these cover numbers less than 10 (because \( \sqrt{79} \approx 8.9 \)).
• 79 wasn’t divisible by any of these, confirming it as a prime number.

Prime numbers like 79 have exactly two distinct natural number divisors: 1 and the number itself.

Number Classification

Classifying numbers into prime and composite categories is foundational in mathematics. It helps understand their properties and how they interact with each other in operations like multiplication and division.
Here's the general idea:

• Prime Numbers: Only two divisors, 1 and itself. Examples include 2, 3, 5, 7, and, as we've seen, 79.
• Composite Numbers: More than two divisors. Examples are 4 (divisible by 1, 2, and 4) and 6 (divisible by 1, 2, 3, and 6).

Remember, the number 1 is unique—it isn’t prime or composite. Number classification aids in understanding fundamental mathematical theories and applications.

Elementary Mathematics

Elementary Mathematics refers to the basics of math taught in early education involving numbers, simple operations, and fundamental concepts like prime and composite numbers.
In elementary math:

• Learning starts by understanding numbers, basic addition, subtraction, division, and multiplication.
• Concepts like divisibility rules and number classification help in grasping complex mathematical theorems later.
• Gaining proficiency in these basics is crucial as they act as a stepping stone to higher mathematical concepts.

By exploring numbers like 79, students can connect these elementary principles to more significant mathematical concepts like factorization, algebra, and beyond.