Problem 81 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The number 101 is a prime number.

Step 1: Check Divisibility

Begin by checking divisibility. To do this, test dividing the number 101 by other prime numbers: 2, 3, 5, 7, 11, 13, 17, and so on, as long as the prime number you are testing as a divisor is less than or equal to the square root of 101, which is approximately 10.05. So you only need to test the primes up to 7.

Step 2: Apply Divisibility Test

Start applying the divisibility test. You will find out that the number 101 is not divisible evenly by any of the primes up to 7.

Step 3: Conclusion

Since no other divisors were found, we can conclude that 101 is a prime number.

Key Concepts

Divisibility Rules

Divisibility rules are handy shortcuts for quickly determining whether a number can be divided by another number without any remainder. By applying these rules, you can save time when searching for factors without doing lengthy calculations. For instance:

• A number is divisible by 2 if it is even (i.e., it ends in 0, 2, 4, 6, or 8).
• It's divisible by 3 if the sum of its digits is divisible by 3.
• A number ends in 0 or 5 if it's divisible by 5.
• For divisibility by 7, double the last digit, subtract it from the rest of the number, and check if the result is divisible by 7.

Using these rules, we can efficiently narrow down potential factors by focusing only on the relevant divisibility checks.

Prime Numbers

Prime numbers are the building blocks of all numbers. These are numbers greater than 1 which have only two divisors, 1 and the number itself. Some essential characteristics include:

• They cannot be divided evenly by any other numbers except 1 and themselves.
• The smallest prime number is 2, which is also the only even prime.
• The next few are 3, 5, 7, 11, 13, and so on.

In the example of 101, after testing divisibility against primes less than or equal to its square root, we found no divisors. Hence, 101 is a prime number. Understanding prime numbers is crucial for topics like prime factorization, where we express a number as the product of its prime factors. But when a number is already a prime, like 101, its prime factorization is just the number itself.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and problem-solving skills. It is the backbone of proving solutions in mathematics, where we explore different methods to reach a conclusion. Here's a breakdown:

• We start by understanding the problem, such as checking if 101 is prime or not.
• We use divisibility checks, concluding that because 101 isn't divisible by any numbers up to around its square root, it must be prime.
• Each step in the reasoning is justified, making sure that our process is sound and leads to a valid conclusion.

By developing mathematical reasoning, students can build confidence and proficiency with concepts such as prime factorization and beyond. Solid reasoning skills enable learners to verify results and avoid common pitfalls in calculations.