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## Are the numbers 343 and 1,000 coprime (prime to each other, relatively prime)?

### 343 and 1,000 are coprime (relatively prime) - if there is no number other than 1 that evenly divides the both numbers (without a remainder) - that is - if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is 1.

## Calculate the greatest (highest) common factor (divisor), gcf (hcf, gcd), of the numbers

### Method 1. The prime factorization:

#### The prime factorization of a number: finding the prime numbers that multiply together to make that number.

#### 343 = 7^{3}

343 is not a prime number, is a composite one.

#### 1,000 = 2^{3} × 5^{3}

1,000 is not a prime number, is a composite one.

#### The numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has only two factors: 1 and itself.

#### A composite number is a natural number that has at least one other factor than 1 and itself.

### >> The prime factorization of numbers

### Calculate the greatest (highest) common factor (divisor), gcf (hcf, gcd):

#### Multiply all the common prime factors of the two numbers, taken by their smallest exponents (powers).

#### But the numbers have no common prime factors.

#### gcf (hcf, gcd) (343; 1,000) = 1

Coprime numbers (prime to each other, relatively prime)

## Coprime numbers (prime to each other, relatively prime) (343; 1,000)? Yes.

The numbers have no common prime factors.

gcf (hcf, gcd) (343; 1,000) = 1

### Method 2. The Euclidean Algorithm:

#### This algorithm involves the process of dividing numbers and calculating the remainders.

#### 'a' and 'b' are the two natural numbers, 'a' >= 'b'.

#### Divide 'a' by 'b' and get the remainder of the operation, 'r'.

#### If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.

#### Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.

#### Step 1. Divide the larger number by the smaller one:

1,000 ÷ 343 = 2 + 314 Step 2. Divide the smaller number by the above operation's remainder:

343 ÷ 314 = 1 + 29 Step 3. Divide the remainder of the step 1 by the remainder of the step 2:

314 ÷ 29 = 10 + 24 Step 4. Divide the remainder of the step 2 by the remainder of the step 3:

29 ÷ 24 = 1 + 5 Step 5. Divide the remainder of the step 3 by the remainder of the step 4:

24 ÷ 5 = 4 + 4 Step 6. Divide the remainder of the step 4 by the remainder of the step 5:

5 ÷ 4 = 1 + 1 Step 7. Divide the remainder of the step 5 by the remainder of the step 6:

4 ÷ 1 = 4 + 0 At this step, the remainder is zero, so we stop:

1 is the number we were looking for - the last non-zero remainder.

This is the greatest (highest) common factor (divisor).

#### gcf (hcf, gcd) (343; 1,000) = 1

### >> The Euclidean Algorithm

## Coprime numbers (prime to each other, relatively prime) (343; 1,000)? Yes.

gcf (hcf, gcd) (343; 1,000) = 1

## The final answer:

(scroll down)

## 343 and 1,000 are coprime (relatively prime) - if there is no number other than 1 that evenly divides the both numbers (without a remainder) - that is - if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is 1. Coprime numbers (prime to each other, relatively prime) (343; 1,000)? Yes.

gcf (hcf, gcd) (343; 1,000) = 1

## The latest 5 pairs of numbers checked on whether they are coprime (prime to each other, relatively prime) or not

Are 343 and 1,000 coprime numbers (relatively prime)? | Nov 24 14:15 UTC (GMT) |

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## Are the two numbers coprime (relatively prime)?

#### Two natural numbers are coprime (relatively prime, prime to each other) - if there is no number that is evenly dividing both numbers (= without a remainder), that is, if their greatest (highest) common factor (divisor), gcf, or hcf, or gcd is 1.

#### Two natural numbers are not relatively prime - if there is at least one number that evenly divides the two numbers, that is, if their greatest common factor, gcf, is not 1.

## Coprime numbers (also called: numbers prime to each other, relatively prime, mutually prime)

- The number "a" and "b" are said to be
**relatively prime, mutually prime, or coprime**if the only positive integer that evenly divides both of them is 1. - The coprime numbers are pairs of (at least two) numbers that do not have any other common factor than 1.
- When the only common factor is 1, then this is also equivalent to their greatest (highest) common factor (divisor) being 1.

**Examples of pairs of coprime numbers:**- The coprime numbers are not necessarily prime numbers themselves, for example 4 and 9 - these two numbers are not prime, they are composite numbers, since 4 = 2 × 2 = 2
^{2}and 9 = 3 × 3 = 3^{2}. But the gcf (4, 9) = 1, so they are coprime, or prime to each other, or relatively prime. - Sometimes, the coprime numbers in a pair are prime numbers themselves, for example (3 and 5), or (7 and 11), (13 and 23).
- Some other times, the numbers that are prime to each other may or may not be prime, for example (5 and 6), (7 and 12), (15 and 23).

**Examples of pairs of numbers that are not coprime:**- 16 and 24 are not coprime, since they are both divisible by 1, 2, 4 and 8 (1, 2, 4 and 8 are their common factors).
- 6 and 10 are not coprime, since they are both divisible by 1 and 2.

**Some properties of the coprime numbers:**- The greatest (highest) common factor (divisor) of two coprime numbers is always 1.
- The least common multiple, LCM, of two coprimes is always their product: LCM (a, b) = a × b.
- The numbers 1 and -1 are the only integers coprime to every integer, for example (1 and 2), (1 and 3), (1 and 4), (1 and 5), (1 and 6), and so on, are pairs of coprime numbers since their greatest (highest) common factor (divisor) is 1.
- The numbers 1 and -1 are the only integers coprime to 0.
- Any two prime numbers are always coprime, for example (2 and 3), (3 and 5), (5 and 7) and so on.
- Any two consecutive numbers are co-prime, for example (1 and 2), (2 and 3), (3 and 4), (4 and 5), (5 and 6), (6 and 7), (7 and 8), (8 and 9), (9 and 10), and so on.
- The sum of two coprime numbers, a + b, is always relatively prime to their product, a × b. For example, 7 and 10 are coprime numbers, 7 + 10 = 17 is relatively prime to 7 × 10 = 70. Another example, 9 and 11 are coprime, and their sum, 9 + 11 = 20 is relatively prime to their product, 9 × 11 = 99.
- A quick way to determine whether two numbers are prime to each other is given by the Euclidean algorithm: The Euclidean Algorithm