Problem 19 - Fundamentals of Mathematics

Question

Find the prime factorization of each whole number. If the number is prime, write "prime." 151

Step-by-Step Solution

Answer

151 is prime.

Step 1: Understanding Prime Factorization

Prime factorization is the process of expressing a number as a product of prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Step 2: Checking for Primality

To determine if 151 is a prime number, we need to check if it has any divisors other than 1 and itself. To do this, attempt dividing 151 by prime numbers less than its square root, approximately 12.3.

Step 3: Testing Divisibility by Smaller Primes

Check the divisibility of 151 by smaller primes such as 2, 3, 5, 7, and 11: - 151 is odd, so not divisible by 2. - Sum of digits of 151 is 7, which is not divisible by 3, so not divisible by 3. - 151 does not end in 0 or 5, so not divisible by 5. - Performing division, 151 ÷ 7 is approximately 21.57 and not a whole number. - Performing division, 151 ÷ 11 is approximately 13.73 and not a whole number.

Step 4: Conclusion and Result

Since 151 is not divisible by any of these smaller prime numbers, it has no divisors other than 1 and itself. Therefore, 151 is a prime number.

Key Concepts

Prime Numbers

Prime numbers are fascinating building blocks in the world of mathematics. They are natural numbers greater than 1 with specific characteristics.
A prime number cannot be divided evenly by any other natural numbers except 1 and itself. This means if you try to divide a prime number by another integer (excluding 1 and itself), you would always get a remainder instead of a whole number.
They are essentially the "atoms" of the number system, used in theories and applications across mathematics, computer science, and more.

• Examples of prime numbers include 2, 3, 5, 7, 11, and 13.
• The number 2 is unique because it is the only even prime number; all other even numbers can be divided by at least 2.
• Prime numbers play a crucial role in prime factorization, which involves expressing numbers as a product of their prime factors, like writing 30 as 2 × 3 × 5.

Divisibility Tests

To effectively determine whether a number is prime or composite, one often uses divisibility tests. These are simple rules that help us identify whether a number can be divided evenly by another.
Divisibility tests allow for quick examinations without laborious calculations.

• For example, a number is divisible by 2 if it is even.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• If a number ends in 0 or 5, it is divisible by 5.

Understanding these rules can save time and effort when determining factors of a number. For testing divisibility against larger primes, simple division may need to be performed to check for whole number results.

Natural Numbers

Natural numbers are the foundation of counting and day-to-day mathematics. They start from 1 and extend infinitely upwards.
Natural numbers include all the positive integers you can count, starting from 1, such as 1, 2, 3, 4, and so on.

• They do not include zero or any negative numbers.
• Natural numbers are used in various mathematical operations, including addition, subtraction, multiplication, and division.
• They form the basis for other types of numbers, such as whole numbers, which include zero, and integers, which include negative numbers.

When dealing with concepts like prime factorization or divisibility, understanding the scope of natural numbers is essential, as these processes are often performed within this set.