Question

Find the number of prime numbers that are less than or equal to 100.

Short answer

There are 25 prime numbers that are less than or equal to 100.

Step-by-step solution

Understand what a Prime Number is

A prime number is a number that has exactly two distinct positive divisors: 1 and the number itself. For instance, the first six prime numbers are 2, 3, 5, 7, 11, and 13.

Set up a range from 2 to 100

Since 1 is not a prime number, we start from 2 and go up to 100.

Find all prime numbers

To find if a number is prime, we check whether it can be divided without a remainder by any number less than it and greater than 1. If it cannot, it means the number is a prime number. By applying this method, the prime numbers less than or equal to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Key concepts

Divisors in Mathematics

In mathematics, divisors are integral to understanding the nature and properties of numbers. When we talk about divisors, we refer to numbers that can be multiplied together to yield another number. Specifically, a divisor of a given number is a number that divides into it without leaving a remainder.

• A number may have multiple divisors: For instance, the number 12 has divisors 1, 2, 3, 4, 6, and 12.
• Prime numbers have a unique property concerning their divisors: they possess exactly two distinct divisors.

For a number to be prime, it must not be divisible by any other number except for 1 and itself.
Understanding divisors helps in identifying prime numbers which play a significant role in various fields of mathematics and algorithmic processes.

Number Theory

Number theory is a branch of pure mathematics devoted primarily to the study of integers. It deals with a wide range of concepts, including prime numbers, divisibility, and the greatest common divisor.

• Prime numbers are foundational in number theory: They are the building blocks of the integers.
• Understanding the distribution and properties of primes is a key area of research.

Number theory aims to solve problems related to the distribution and patterns of integers. The study of prime numbers answers questions like how many there are under a certain limit, such as within the range from 2 to 100. Knowing these enables deeper mathematical insights and informs coding and encryption in technology.
The historical and practical significance of number theory is immense, providing tools and methods to approach various mathematical problems.

Mathematical Algorithms

Mathematical algorithms are step-by-step procedures or formulas for solving mathematical problems. These algorithms are crucial for efficiently calculating outcomes and making processes like identifying prime numbers viable.

• Prime identification: An algorithm might involve checking divisibility of a number, like testing divisors from 2 up to the square root of the number in question.
• Efficiency matters: Directly checking all numbers is inefficient, hence algorithms apply smarter checks, reducing time complexity.

Consider the Sieve of Eratosthenes, a classic algorithm for finding all prime numbers up to a specified integer. It works by iteratively marking the multiples of each prime starting from 2. Such algorithms are foundational in computer science, cryptography, and various applied mathematics fields.
They ensure that calculations are optimized and effective, demonstrating the power of mathematical processes.