Problem 62 - Fundamentals of Mathematics

Question

Write each number as a product of prime factors. 921

Step-by-Step Solution

Answer

921 = 3 × 307.

Step 1: Check for Divisibility by 2

To find the prime factors, we begin with 2, the smallest prime number. Since 921 is odd, it’s not divisible by 2.

Step 2: Check for Divisibility by 3

Add the digits of 921: 9 + 2 + 1 = 12. Since 12 is divisible by 3, 921 is divisible by 3. Dividing, we get 921 ÷ 3 = 307.

Step 3: Verify if 307 is a Prime Number or Check for Divisibility by Small Primes

Check if 307 is divisible by the next smallest primes (5, 7, 11, 13, etc.). It ends with 7, so it’s not divisible by 5. Divisibility rules for 7, 11, and 13 do not apply, indicating 307 is not divisible by these. Upon checking, 307 is confirmed as a prime number.

Step 4: Write 921 as a Product of Prime Factors

Since 921 = 3 × 307, and both 3 and 307 are prime numbers, we conclude that the prime factorization of 921 is 3 × 307.

Key Concepts

Prime Numbers

Prime numbers are the building blocks of mathematics. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it can't be divided evenly by any other number. Here are some examples:

• 2
• 3
• 5
• 7
• 11

The importance of prime numbers is enormous in mathematics because every number is either a prime or a product of primes. Understanding prime numbers helps solve various mathematical problems, like finding prime factors or simplifying expressions. They are crucial in fields like cryptography, where they are used to create keys for encrypting data.

Divisibility Rules

Divisibility rules help us quickly determine if a number can be divided by another without actually doing the long division. They offer shortcuts, especially when dealing with large numbers. Here are some important rules:

• **Divisible by 2**: A number is divisible by 2 if it is even (i.e., it ends in 0, 2, 4, 6, or 8).
• **Divisible by 3**: Add up all the digits of the number. If the sum is divisible by 3, then so is the original number.
• **Divisible by 5**: A number is divisible by 5 if it ends in 0 or 5.

For the exercise, noticing that 921 is not divisible by 2 helped us skip testing that factor. Summing the digits to see 12 is divisible by 3 guided us successfully to our first prime factor, 3. These rules streamline the process, making tasks like prime factorization easier.

Mathematical Problem-Solving

Solving mathematical problems requires a systematic approach. To find the prime factorization of 921, we followed several crucial problem-solving steps:

• **Identify the Problem**: We needed the prime factors of the number 921.
• **Plan the Approach**: Starting from the smallest prime number and applying divisibility tests helped us create a strategy for tackling the problem.
• **Execute the Plan**: We checked divisibility rules for small primes until we found factors (3 and 307).
• **Verify the Solution**: Confirming 307 as a prime number ensured our factorization was complete.

This disciplined method not only aids in solving specific exercises but also builds a broader understanding and confidence in tackling mathematical challenges.

Product of Prime Factors

Writing a number as a product of its prime factors is known as prime factorization. This technique breaks the number down into the prime numbers that multiply together to reach the original number. For example, the prime factorization of 921 is expressed as:\[921 = 3 \times 307\] Here's why it's beneficial:

• **Simplification**: It simplifies expressions, allowing for easier manipulation in algebra and calculus.
• **Lowest Common Multiple & Greatest Common Divisor**: Prime factorization is useful in computing these for numbers, which is valuable for solving fraction-related problems.
• **Cryptography**: Primes help in generating secure cryptographic keys, crucial for data protection.

Understanding how to express numbers as a product of prime factors is a fundamental skill in mathematics.