Problem 97 - Algebra for College Students

Question

Is \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11+7\) a prime or a composite number? Defend your answer.

Step-by-Step Solution

Answer

2317 is a composite number because it is divisible by 7.

Step 1: Prime Number Definition

First, recall that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

Step 2: Calculate the Expression

Calculate the expression given: \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 + 7\). First, find the product of the prime numbers: \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310\). Adding 7 gives \(2310 + 7 = 2317\).

Step 3: Check Divisibility by 7

Since the original expression added 7 to a multiple of 7, check if 2317 is divisible by 7. Divide 2310 by 7 and verify that it gives a whole number. Therefore, 2317 is divisible by 7.

Step 4: Conclude with Prime or Composite

Since 2317 has a divisor other than 1 and itself (it is divisible by 7), 2317 is a composite number.

Key Concepts

Composite Numbers

A number is called composite if it has more than two distinct positive divisors. This means that, unlike prime numbers, composite numbers can be divided exactly by numbers other than 1 and themselves. When working with composite numbers, you can think of them as numbers that can be "broken down" into smaller numbers. For example, 2317 is a composite number.

• It can be divided not only by 1 and 2317, but also by 7.
• Having divisors other than 1 and itself confirms its composite nature.

Understanding whether a number is composite helps in many mathematical areas, including simplifying fractions and finding common denominators.

Divisibility

Divisibility is a concept that helps to determine if one number can be divided by another without leaving any remainder. For example, a number is divisible by 7 if, after division, the result is a whole number.

• To test if a number like 2317 is divisible by 7, you perform division.
• If 2317 divided by 7 equals an integer, then it is divisible by 7.

In this exercise, 2317 is shown to be divisible by 7, making divisibility a useful tool for understanding why a number is composite. It's a fundamental part of number theory and helps with tasks like simplifying expressions and verifying number properties.

Prime Factorization

Prime factorization involves expressing a composite number as a product of prime numbers. This process breaks down composite numbers into their basic building blocks.

• Start by dividing the number by the smallest prime number.
• Continue dividing the result by prime numbers until you reach a prime number.

For example, while the initial part of the exercise dealt with prime numbers up to 11, the resulting composite number, 2317, can be further factorized through divisibility by 7. Knowing how to perform prime factorization is quite practical—it simplifies solving mathematical problems and understanding number structures.