Problem 96 - Algebra 1: Concepts and Skills

Question

Write the prime factorization of the number if it is not a prime number. If a number is prime, write prime. $$ 101 $$

Step-by-Step Solution

Answer

The number 101 is a prime number. Therefore, its prime factorization is simply 101.

Step 1: Understanding Prime Numbers

Remember, prime numbers are numbers that have only two distinct positive divisors: 1 and the number itself. For example, the first six prime numbers are 2, 3, 5, 7, 11, and 13.

Step 2: Identify if the Number Is a Prime Number

Check if 101 can be divided evenly by any prime number less than its square root. The square root of 101 is approximately 10.05, so we'll check the prime numbers 2 (the only even prime number), 3, 5, and 7. Since 101 is an odd number and cannot be divided evenly by 2, and is not divisible by 3, 5, or 7, we can conclude 101 is a prime number.

Step 3: Conclude and Write Down the Result

Since the number 101 is a prime number (it has no divisors apart from 1 and itself), its prime factorization is simply 101.

Key Concepts

Prime Numbers

Prime numbers are quite special in mathematics. They are numbers greater than 1 that have no other divisors but 1 and themselves. This means you cannot divide a prime number evenly by any number other than 1 and itself without getting a remainder. Prime numbers serve as the building blocks for all whole numbers because every whole number is a product of prime numbers. For example, number 6 can be broken down into the prime numbers 2 and 3, since 2 \( \times \) 3 = 6.

Understanding this concept is essential because it allows us to express any whole number uniquely as a product of primes, known as its prime factorization.

• This helps greatly in mathematics, particularly in areas like cryptography and number theory.
• It is important to remember that the number 1 is not considered a prime number, as it only has one divisor, which is itself.

Divisors

Divisors are numbers that divide another number completely without leaving a remainder. Every number has divisors. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. You use divisors to determine if a number is prime or not.

If a number has only two divisors—1 and itself—it’s a prime number. But if it has more than two, then it's composite. In this context, the number 101 was tested for divisibility by several smaller prime numbers. Since no number other than 1 and 101 itself could divide 101 with no remainder, we concluded that 101 is prime.

• This process confirms that divisors are crucial in many mathematical processes, such as finding the greatest common divisor.
• Understanding how to find divisors aids in simplifying fractions and solving problems involving ratios.

Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 \( \times \) 5 = 25. When determining if a number is prime, we often use the square root method.

To check if a number is prime, we see if it has divisors other than 1 and itself. This is efficiently done by checking divisibility with prime numbers up to the square root of the number in question. For 101, its square root is approximately 10.05. So, we only needed to check divisibility with the prime numbers less than 10.05, such as 2, 3, 5, and 7.

• Using the square root reduces the number of calculations needed, thus simplifying the process of prime checking.
• Square roots are used not only in prime factorization but also in geometry, physics, and various fields requiring calculations involving areas and vectors.