# Mathematical Operations With Prime Numbers

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

 What is the prime factorization of the composite number 136,518,425? Sep 22 09:26 UTC (GMT) What is the prime factorization of the composite number 514,596,662? Sep 22 09:26 UTC (GMT) What is the prime factorization of the composite number 1,772,690,210? Sep 22 09:26 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

 What is the greatest (highest) common factor (divisor) of the numbers 6,104 and 4,190? How to calculate the GCF (HCF, GCD)? Sep 22 09:26 UTC (GMT) What is the greatest (highest) common factor (divisor) of the numbers 6,117 and 4,204? How to calculate the GCF (HCF, GCD)? Sep 22 09:26 UTC (GMT) What is the greatest (highest) common factor (divisor) of the numbers 16 and 5,066? How to calculate the GCF (HCF, GCD)? Sep 22 09:25 UTC (GMT) The greatest (highest) common factor (divisor), gcf (hcf, gcd): the list of all the calculations

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

 What is the least (the lowest) common multiple, LCM, of the numbers 999,999,999,949 and 1,998 and how to calculate it? Sep 22 09:26 UTC (GMT) What is the least (the lowest) common multiple, LCM, of the numbers 1,000,000,017 and 1,000,000,014 and how to calculate it? Sep 22 09:26 UTC (GMT) What is the least (the lowest) common multiple, LCM, of the numbers 9 and 4,112 and how to calculate it? Sep 22 09:26 UTC (GMT) The least (the lowest) 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)

 Completely reduce (simplify) the fraction 3,024 / 738 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Sep 22 09:26 UTC (GMT) Completely reduce (simplify) the fraction 15,443 / 42 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Sep 22 09:26 UTC (GMT) Completely reduce (simplify) the fraction 813,754 / 22 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Sep 22 09:26 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 the number 88,046 divisible by 37? Could 88,046 be evenly divided by 37? Does the first number contain all the prime factors of the second? Sep 22 09:26 UTC (GMT) Is the number 422 divisible by 446? Could 422 be evenly divided by 446? Does the first number contain all the prime factors of the second? Sep 22 09:26 UTC (GMT) Is the number 972 divisible by 263? Could 972 be evenly divided by 263? Does the first number contain all the prime factors of the second? Sep 22 09:26 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

 What are all the proper, improper and prime factors (all the divisors) of the number 494,851? How to calculate them? Sep 22 09:26 UTC (GMT) What are all the proper, improper and prime factors (all the divisors) of the number 23,052,006? How to calculate them? Sep 22 09:26 UTC (GMT) What are all the proper, improper and prime factors (all the divisors) of the number 1,374,600? How to calculate them? Sep 22 09:26 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 the two numbers 6,640 and 9,907 coprime (relatively prime, prime to each other) or not? Sep 22 09:26 UTC (GMT) Are the two numbers 55 and 1,701,700 coprime (relatively prime, prime to each other) or not? Sep 22 09:26 UTC (GMT) Are the two numbers 3,216 and 65 coprime (relatively prime, prime to each other) or not? Sep 22 09:26 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 operations on numbers' parity: even or odd numbers?

 Numbers parity: Is 1,684,982 an even or an odd number? Sep 22 09:26 UTC (GMT) Numbers parity: Is 8,829,848,788 an even or an odd number? Sep 22 09:26 UTC (GMT) Numbers parity: Is 35,825,065 an even or an odd number? Sep 22 09:26 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.