Proving/Disproving: Coprimes & (an + bn + cn ) / abc

Thread starter mahmudarif
Start date Aug 13, 2012

In summary, three numbers are coprime if they do not have any common factors other than 1 and their greatest common divisor is 1. To prove that three numbers are coprime, you can use the Euclidean algorithm or check their prime factorizations. If two numbers are not coprime, then any combination of them with a third number will also not be coprime. The expression (an + bn + cn) / abc can be used to test if three numbers are coprime and can also be used to prove or disprove the claim.

Aug 13, 2012

mahmudarif

if a, b, c (integers) are coprimes then (aⁿ + bⁿ + cⁿ) / abc can't be a number that is not a fraction (integer). Any idea to prove/disprove it will be highly appreciated.

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Aug 13, 2012

Mandlebra

If the sum is divisibleby abc what does that say about the terms

Aug 15, 2012

haruspex

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2³+3³+7³

1. What does it mean for three numbers to be coprime?

Three numbers are coprime if they do not have any common factors other than 1. In other words, their greatest common divisor is 1. For example, 4, 9, and 13 are coprime because the only positive integer that divides all three of them is 1.

2. How can I prove that three numbers are coprime?

To prove that three numbers are coprime, you can use the Euclidean algorithm to find their greatest common divisor. If the greatest common divisor is 1, then the numbers are coprime. Alternatively, you can also check if the prime factorization of each number has no common factors.

3. Can three numbers be coprime if two of them are not coprime?

No, if two numbers are not coprime, then their greatest common divisor is greater than 1. This means that any combination of these two numbers and a third number will also have a greatest common divisor greater than 1, and therefore, the three numbers cannot be coprime.

4. How does the expression (an + bn + cn) / abc relate to coprime numbers?

The expression (an + bn + cn) / abc is used to test if three numbers (a, b, and c) are coprime. If the expression simplifies to an integer, then the numbers are coprime. If the expression does not simplify to an integer, then the numbers are not coprime.

5. Can the expression (an + bn + cn) / abc be used to prove or disprove that three numbers are coprime?

Yes, the expression (an + bn + cn) / abc can be used as a proof or disproof of coprime numbers. If the expression simplifies to an integer, then it proves that the three numbers are coprime. If the expression does not simplify to an integer, then it disproves the claim that the numbers are coprime.