Problem 57 - Basic College Mathematics

Question

Find the prime factorization of each number. $$ 100 $$

Step-by-Step Solution

Answer

The prime factorization of 100 is 2 × 2 × 5 × 5.

Step 1 - Start with the smallest prime number

Begin by dividing the number 100 by the smallest prime number, which is 2. 100 ÷ 2 = 50

Step 2 - Continue with the same prime number

Since 50 is still divisible by 2, divide it by 2 again. 50 ÷ 2 = 25

Step 3 - Move to the next prime number

Now 25 is not divisible by 2, so move to the next smallest prime number, which is 3. Since 25 is not divisible by 3, try the next prime number, which is 5. 25 ÷ 5 = 5

Step 4 - Continue with prime number 5

Since 5 is still divisible by 5, divide it by 5. 5 ÷ 5 = 1

Step 5 - Write down the prime factors

We have finished the process when we reach 1. Now, list all the prime numbers used in the divisions: 2, 2, 5, and 5. Therefore, the prime factorization of 100 is 2 × 2 × 5 × 5.

Key Concepts

smallest prime number

The smallest prime number is 2. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. This means 2 can only be divided by 1 and 2 without leaving a remainder. Understanding that 2 is the smallest prime number is essential in factorization because it is the first number we check when breaking down a larger number.

divisibility

Divisibility refers to the ability for one number to be divided by another without leaving a remainder. This concept is crucial when performing prime factorization. For example, in the problem above, 100 is divisible by 2 because 100 ÷ 2 = 50 with no remainder. We continue this process by testing divisibility with the same or next smallest prime numbers, like checking if 50 can be divided by 2 again, and so on.

step by step solution

Prime factorization can sometimes seem complex, but breaking it down into clear, manageable steps simplifies the task. Let's rehearse the process with 100:

1. Start by dividing by the smallest prime number, which is 2.
100 ÷ 2 = 50
2. Since 50 is still divisible by 2, divide again.
50 ÷ 2 = 25
3. Now, 25 isn't divisible by 2. We try the next prime numbers (3, and then 5).
25 ÷ 5 = 5
4. Finally, divide once more by 5.
5 ÷ 5 = 1

With 1 remaining, list all the primes used: 2, 2, 5, and 5. Thus, 100 = 2 × 2 × 5 × 5.

prime factors

Prime factors are the prime numbers that multiply together to make the original number. For the number 100, these factors are 2 and 5. When we broke down 100 in the steps above, we ended up with 2, 2, 5, and 5.

Queue prime factorization:
100 = 2 × 2 × 5 × 5
Understanding prime factors helps not only in simplification but also in understanding the fundamental building blocks of any given number.