Prime Numbers 700 to 800

A prime number is a natural number with no positive factors other than 1 and the number itself. A prime number can only be evenly divisible by 1 and the number itself. Here are some basic properties of prime numbers:

• Every number greater than 1 is divisible by at least one prime number.

• Two prime numbers are always relatively prime to each other.

• Every even positive integer greater than 2 can be written as the sum of two prime numbers.

• Every composite number can be uniquely factored into prime factors.

• Except for 2, all prime numbers are odd; 2 is the only even prime number.

A prime number chart is a table showing the prime numbers in increasing order. The chart includes all the prime numbers up to a certain limit for identifying the prime numbers within a range.

For kids, it will be less difficult to understand the prime numbers through the chart. The significance of this prime number chart is used in different fields like the foundation of mathematics and the fundamental theorem of arithmetic.

The list of all prime numbers from 700 to 800 provides a comprehensive view of numbers in this range that can only be divided by 1 and the number itself. The prime numbers in the range of 700 to 800 include 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797.

Prime numbers and odd numbers are the numbers that are only divisible by 1 and the number itself. They cannot be evenly divisible by 2 or other numbers. 2 is the only even prime number, which divides all the non-prime numbers. Therefore, except for 2, all prime numbers are considered as the set of odd numbers.

Prime numbers are a set of natural numbers that can only be divided by 1 and the number itself. Here are two important ways to find whether a number is prime or not.

By Divisibility Method:

To find whether a number is prime or not, we use the divisibility method to check. If a number is divisible by 2, 3, or 5, then it will result in a non-prime number. Prime numbers are only divisible by 1 and itself, so if a number is divisible by the number itself and 1, it is a prime number.

For example: To check whether 727 is a prime number,

Step 1: 727 ÷ 2 = 363.5 (remainder ≠ 0)

Step 2: 727 ÷ 3 = 242.33 (remainder ≠ 0)

Step 3: 727 ÷ 5 = 145.4 (remainder ≠ 0)

Since no divisors are found, 727 is a prime number.

By Prime Factorization Method:

The prime factorization method is the process of breaking down a composite number into the product of its prime factors. The method of prime factorization helps to identify the prime numbers up to 800 by building the smallest blocks of any given number.

For example: The prime factorization of 800: Let's break it down into the smallest prime numbers until it can’t divide anymore.

Step 1: 800 ÷ 2 = 400

Step 2: Now, we divide 400, 400 ÷ 2 = 200

Step 3: Now take 200, 200 ÷ 2 = 100

Step 4: Take 100, 100 ÷ 2 = 50

Step 5: Take 50, 50 ÷ 2 = 25

Step 6: At last, take 25, 25 ÷ 5 = 5

Step 7: Finally, take 5, 5 ÷ 5 = 1 (since 5 is a prime number, and dividing by 5 gives 1)

Therefore, the prime factorization of 800 is: 800 = 2⁵ × 5².

Rule 1: Divisibility Check: Prime numbers are natural numbers that are greater than 1 and have no divisors other than 1 and the number itself. In the divisibility check rule, we check whether the prime number is divisible by 2, 3, 5, 7, 11, 13, 17, 19, or 23. If it's divisible by these numbers, then it's not a prime number.

Rule 2: Prime Factorization: In this prime factorization method, we break down all the numbers into their prime factors, showing them as the product of prime numbers.

Rule 3: Sieve of Eratosthenes Method: The method, sieve of Eratosthenes, is an ancient algorithm used to find all prime numbers up to a given limit. First, we list all the numbers from 700 to 800. Then start with the first prime number, 701. Mark all the multiples of 701 as non-prime.

Repeat the process for the next unmarked prime number and continue until you reach the square root of 800, approximately 28.28. The remaining unmarked numbers are the prime numbers.

Tips and Tricks for Prime Numbers 700 to 800

Use common shortcuts to memorize the prime numbers. 701, 709, 719, 727, 733, 739, 743, 751, and 757 use these numbers as reference. Practice using the method of Sieve of Eratosthenes efficiently. Numbers like 704, 708, 716, 728, and 736 are never meant to be prime. Knowing the common powers of numbers helps in avoiding unnecessary checks.

• Prime numbers: The natural numbers which are greater than 1 and that are divisible by 1 and the number itself. For example: 701, 709, 719, 727, and so on.

• Odd numbers: The numbers that are not divisible by 2 are called odd numbers. All prime numbers except 2 are odd. For example: 701, 709, 719, and so on.

• Composite numbers: Composite numbers are non-prime numbers that have more than 2 factors. For example, 800 is a composite number, and it is divisible by 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 160, 200, 400, and 800.

• Divisibility: A method to check if a number can be divided by another without a remainder. For example, 720 ÷ 5 leaves a remainder, so 720 is not divisible by 5.

• Sieve of Eratosthenes: An ancient algorithm to discover all prime numbers up to a specified integer. It removes multiples of each prime number starting with 2.