Problem 59 - Fundamentals of Mathematics

Question

Write each number as a product of prime factors. 700

Step-by-Step Solution

Answer

The prime factors of 700 are \(2^2 \times 5^2 \times 7\).

Step 1: Identify the Smallest Prime Factor

To find the prime factors of 700, we begin by identifying the smallest prime factor that divides into 700. Since 700 is an even number, we divide by 2. \(700 \div 2 = 350\). Therefore, the smallest prime factor is 2.

Step 2: Continue Factoring 350

Next, we take the result from Step 1, which is 350, and find its smallest prime factor. Again, since 350 is even, divide by 2. \(350 \div 2 = 175\). Thus, another prime factor is 2.

Step 3: Factor 175

Now we factor 175. Since 175 is not even, we check the next smallest prime number, which is 3. 175 is not divisible by 3, so we try the next prime, 5. \(175 \div 5 = 35\). So, 5 is a prime factor.

Step 4: Continue with 35

For 35, we still look for its smallest prime factor. Starting with 5 (since 35 ends in 5), 35 is divisible by 5. \(35 \div 5 = 7\). So another prime factor is 5.

Step 5: Finalize with 7

The result from Step 4 is 7. Since 7 is already a prime number, we stop the factorization here. Thus, 7 is part of the prime factorization of 700.

Step 6: Write the Product of Prime Factors

Collecting all the prime factors identified, we write 700 as a product of its prime factors: \(700 = 2^2 \times 5^2 \times 7\).

Key Concepts

Prime Numbers

Prime numbers are the building blocks for all whole numbers. They are numbers greater than 1 that can only be divided evenly (without a remainder) by 1 and themselves. This means when you attempt to divide a prime number by any other number, you will end up with a decimal or fraction rather than a whole number. Here are a few things to remember about prime numbers:

• The first five prime numbers are 2, 3, 5, 7, and 11.
• 2 is the only even prime number because every other even number can be divided by 2.
• Primes are integral to the process of breaking down numbers into their basic components, or prime factorization.

Understanding prime numbers helps you in identifying factors easily since once a prime factorization is achieved, these prime numbers can be used to reconstruct any number they build.

Division

Division is one of the fundamental operations in arithmetic, where you divide a number into equal parts. To find the prime factorization of a number, division is extensively used to break it down into its smallest prime factors. Here’s how division helps in prime factorization:

• Identify the smallest prime number that divides the given number without a remainder.
• Repeat the process with the quotient obtained after each division.
• Stop when the quotient itself is a prime number.

For example, in the factorization of 700, we divided it firstly by 2 since it was even, then continued dividing the resulting quotients until all were prime numbers.
Diving deeper into division helps develop efficiency in solving factorization problems and understanding relationships between numbers.

Even Numbers

Even numbers are numbers that are divisible by 2. They always end in 0, 2, 4, 6, or 8 when written in base ten. Being able to identify even numbers is crucial in the process of prime factorization because it allows you to quickly find and use 2 as a factor.
Here’s what’s helpful to know about even numbers during factorization:

• The first division step often involves 2 if the number is even.
• Continuing with the division by 2 helps simplify the number quickly in many cases.
• Once an even number is minimized entirely by 2, it becomes easier to test for other prime factors.

In the example of 700, we first identified it as even, dividing first by 2 to simplify the factorization process.
Recognizing even numbers offers a head start in prime factorization, helping you to save time by targeting smaller, more manageable numbers.

Multiplication

Multiplication is the process of adding a number to itself a certain number of times. In the context of prime factorization, multiplication helps to verify and understand the composition of a number.
Once all prime factors are identified, they can be multiplied together to reconstruct the original number.
Here’s how multiplication is utilized:

• It helps confirm the correctness of the factorization by multiplying all the prime factors together.
• It allows writing numbers in a compact form like the exponent notation, such as expressing repeated factors succinctly as powers.
• This facilitates the examination of number properties, like divisibility and the calculation of least common multiples and greatest common factors.

Using multiplication, we verify the factorization of 700 by multiplying its prime factors: \[700 = 2^2 \times 5^2 \times 7\]. By confirming the results, we ensure the correctness of our division work and reinforce our understanding of how numbers can be broken down and rebuilt back up.