# Mathematical Operations With Prime Numbers

## Prime or composite numbers? The last 3 numbers on which the prime factorization has been performed

 Is 47,843 a prime number or not? Sep 29 07:58 UTC (GMT) The prime factorization of 196 = ? Sep 29 07:58 UTC (GMT) The prime factorization of 166,377 = ? Sep 29 07:58 UTC (GMT) The list of numbers that were checked on whether they are prime or not. The prime factorization operations of the composite numbers.

## The greatest (highest) common factor (divisor), gcf (hcf, gcd): the latest 3 calculated values

 The gcf, hcf, gcd (5,536 and 3,868) = ? Sep 29 07:58 UTC (GMT) The gcf, hcf, gcd (4,236 and 194) = ? Sep 29 07:58 UTC (GMT) The gcf, hcf, gcd (592 and 300) = ? Sep 29 07:58 UTC (GMT) The greatest (highest) common factor (divisor), gcf (hcf, gcd): the list of all the calculations

## The least common multiple, LCM: the latest 3 calculated values

 The LCM of 56 and 104 = ? Sep 29 07:59 UTC (GMT) The LCM of 6,370 and 72 = ? Sep 29 07:58 UTC (GMT) The LCM of 2,324,322 and 258,258 = ? Sep 29 07:58 UTC (GMT) The least common multiple, LCM: the list of all the operations

## The latest 3 fractions that have been fully reduced (simplified) to their lowest terms (to their simplest form, the smallest possible numerator and denominator)

 Reduce (simplify) the fraction: 259/5 = ? Sep 29 07:59 UTC (GMT) Reduce (simplify) the fraction: 12/27 = ? Sep 29 07:58 UTC (GMT) Reduce (simplify) the fraction: 10,939/9,851 = ? Sep 29 07:58 UTC (GMT) The list of all the fractions that were fully reduced (simplified) to their lowest terms (to their simplest form), the smallest possible numerator and denominator

## Divisibility: the latest 3 pairs of numbers checked on whether they are divisible or not

 Is 2,194 divisible by 273? Sep 29 07:58 UTC (GMT) Is 101,342 divisible by 4? Sep 29 07:58 UTC (GMT) Is 5,664 divisible by 5? Sep 29 07:58 UTC (GMT) The list of all the pairs of numbers that were checked on whether they are divisible or not

## The latest 3 sets of calculated factors (divisors): of one number or the common factors of two numbers

 The common factors (divisors) of 5,664,331 and 0 = ? Sep 29 07:59 UTC (GMT) The factors (divisors) of 2,062,632 = ? Sep 29 07:59 UTC (GMT) The common factors (divisors) of 379,364,860 and 0 = ? Sep 29 07:59 UTC (GMT) The list of all the calculated factors (divisors) of one or two numbers

## The latest 3 pairs of numbers checked on whether they are coprime (prime to each other, relatively prime) or not

 Are 1,462 and 8,993 coprime numbers (relatively prime)? Sep 29 07:58 UTC (GMT) Are 6,130 and 6 coprime numbers (relatively prime)? Sep 29 07:58 UTC (GMT) Are 15 and 16 coprime numbers (relatively prime)? Sep 29 07:58 UTC (GMT) All the pairs of numbers that were checked on whether they are coprime (prime to each other, relatively prime) or not

## The latest 3 checked on numbers: even or odd number?

 Is 1,608,452 an even or an odd number? Sep 29 07:58 UTC (GMT) Is 10,596 an even or an odd number? Sep 29 07:58 UTC (GMT) Is 460,173 an even or an odd number? Sep 29 07:57 UTC (GMT) The list of all the checked on numbers: is it an even or an odd number?

## 1. Prime numbers. 2. The fundamental theorem of arithmetic. 3. Composite numbers. 4. Remarks

• ### 1. Prime numbers

• A prime number is a natural number, larger than 1, which is evenly dividing (= without a remainder) only by 1 and itself.
• Any "m" prime number has only two divisors (two factors): the number itself, "m", and the number 1.
• Examples of prime numbers:
• 1 is not considered a prime number, so the first prime number is 2 (the prime numbers list is starting with the number 2).
• 2 is divisible only by 2 and 1, so 2 is a prime number.
• 3 is divisible only by 3 and 1, so 3 is a prime number.
• 5 is divisible only by 5 and 1, so 5 is a prime number.
• 13 is divisible only by 13 and 1, so 13 is a prime number.
• ### 2. The fundamental theorem of arithmetic

• The fundamental theorem of arithmetic says that every natural number larger than 1 can be written as a product of one or more prime numbers in a way that is unique, except for the order of the prime factors.
• Why is 1 not considered a prime number? If 1 were considered a prime number, then the prime factorization of the number 15, for example, could be either: 15 = 3 × 5 or 15 = 1 × 3 × 5. These two representations would have been considered two different prime factorizations of the same number, 15, so the statement of the fundamental theorem would no longer be true.
• ### 3. Composite numbers

• A composite number is a natural number that has at least one positive divisor (factor) other than 1 and the number itself.
• A composite number is also any number larger than 1 that is not a prime number.
• The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
• Examples of composite numbers:
• 4 is divisible by 4, 2 and 1, so 4 is not a prime number, it is a composite number. The prime factorization of 4 = 2 × 2 = 22
• First Note: The second part of the prime factorization of 4 is written by using powers and exponents and it is called a condensed writing of the prime factorization.
• Second Note: 23 = 2 × 2 × 2 = 8. 2 is called the base and 3 is the exponent. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power. We sometimes say that the number 2 was raised to the power of 3.
• 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime number, it is a composite number. The prime factorization of 6 = 2 × 3
• 8 is divisible by 8, 4, 2 and 1, so 8 is not a prime number, it's a composite number. The prime factorization is 8 = 23
• 9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number. Its prime factorization: 9 = 32
• ### 4. Remarks on the prime numbers

• The list of the first prime numbers, up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
• The prime numbers are the basic building blocks of all the numbers, taking into consideration that every number can be written as a product of one or more primes. Every composite number can be written as a product of at least two prime numbers.
• EUCLID (300 B.C.) proved that as the set of natural or integer numbers is infinite, also the the set of prime numbers is infinite, with no largest prime number.
• There is no known simple formula that sets apart all of the prime numbers from the composite ones.