Are 2,525 and 2,178 coprime (prime to each other, relatively prime)?
2,525 and 2,178 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.
2,525 = 52 × 101
2,525 is not a prime number, is a composite one.
2,178 = 2 × 32 × 112
2,178 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.
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) (2,525; 2,178) = 1
Coprime numbers (prime to each other, relatively prime)
Coprime numbers (prime to each other, relatively prime) (2,525; 2,178)? Yes.
The numbers have no common prime factors.
gcf (hcf, gcd) (2,178; 2,525) = 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:
2,525 ÷ 2,178 = 1 + 347
Step 2. Divide the smaller number by the above operation's remainder:
2,178 ÷ 347 = 6 + 96
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
347 ÷ 96 = 3 + 59
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
96 ÷ 59 = 1 + 37
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
59 ÷ 37 = 1 + 22
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
37 ÷ 22 = 1 + 15
Step 7. Divide the remainder of the step 5 by the remainder of the step 6:
22 ÷ 15 = 1 + 7
Step 8. Divide the remainder of the step 6 by the remainder of the step 7:
15 ÷ 7 = 2 + 1
Step 9. Divide the remainder of the step 7 by the remainder of the step 8:
7 ÷ 1 = 7 + 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) (2,525; 2,178) = 1
Coprime numbers (prime to each other, relatively prime) (2,525; 2,178)? Yes.
gcf (hcf, gcd) (2,178; 2,525) = 1
The final answer:
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2,525 and 2,178 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) (2,525; 2,178)? Yes.
gcf (hcf, gcd) (2,525; 2,178) = 1
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