Greatest Common Divisor of Four Distinct Positive Integers

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In summary, the conversation discusses finding a set of four distinct positive integers that satisfy the condition of having a greatest common divisor of 6 for all six pairs of elements in the set. The suggested solution is to use numbers in the form 6n, where n is a distinct prime or 1.
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knowLittle
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Homework Statement


Give an example of a set S of four (distinct) positive integers such that the greatest common divisor of all
six pairs of elements of S is 6.

Homework Equations


The Attempt at a Solution



Can I say that my numbers are in the form?
6
12
18
30
Is this ok?
 
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  • #2
Are you sure you have the question worded correctly? The title says four pairs, the question refers to six.
 
  • #3
You don't need to skip 24. Using 6 as your first number will guarantee that the gcd is 6 as long as all of your other integers are of the form 6n where n is an integer.
 
  • #4
Nascent,
4 choose 2 at a time is 6 possible pairs of gcd.

Mentallic,
If I used gcd(12, 24) = 12 and it doesn't satisfy the conditions.
 
  • #5
Ahh that makes a lot more sense now. Then yes, what you've done is correct. It just needs to be of the form

[tex]6p_1, 6p_2, 6p_3, 6p_4[/tex]

where pn is a distinct prime or 1.
 
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Thanks :>
 

FAQ: Greatest Common Divisor of Four Distinct Positive Integers

What is GCD?

GCD stands for Greatest Common Divisor. It is the largest positive integer that divides evenly into all of the given numbers.

What is the purpose of finding the GCD of four different pairs?

The GCD of four different pairs is used to find the largest number that can divide evenly into all sets of four numbers. It is commonly used in mathematics and computer programming to simplify fractions and find the lowest common denominator.

How is the GCD of four different pairs calculated?

The GCD of four different pairs is calculated by finding the common factors of all four pairs and selecting the largest one. This can be done by listing out all the factors of each number and finding the ones that are common to all four numbers. Alternatively, the Euclidean algorithm can be used to find the GCD more efficiently.

What happens if one or more of the numbers in the four pairs is 0?

If one or more of the numbers is 0, the GCD of the four different pairs will be the non-zero number. This is because 0 can be divided by any number without a remainder, so it does not affect the GCD calculation.

Can the GCD of four different pairs be negative?

No, the GCD of four different pairs is always a positive integer. This is because negative numbers can also be divided by positive numbers without a remainder, so the negative sign does not affect the GCD calculation.

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