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Relatively Prime Numbers

What is Relatively Prime?

By definition, two numbers are relatively prime if and only if the greatest common divisor of both numbers is 1.

A Few Loose Ends

Question: Are there numbers greater than x that are relatively prime to x ?

Sure. Relatively prime is a pairwise relationship. It takes two numbers to be relatively prime. Certainly 15 is relatively prime to 32, but also 32 is relatively prime to 15.

Question: Is there a number that is relatively prime to every postive number?

Yup! That number is 1. Remember, by definition, two numbers are relatively prime if their greatest common divisor is 1. The GCD of 1 and x for all x is 1.

Question: Is there any other use of relatively prime? Oh yeah! The one important use that comes to mind quickly is finding out how many proper fractions are reduced to lowest terms with a given denominator. If you have a fraction that is not reduced to lowest terms, then there is some number (other than 1) that will divide both the numerator and denominator evenly. That number, then, is a common divisor (from which you can find a greatest common divisor, but the gcd won’t be 1). So, the problems “How many positive numbers less than or equal to x are relatively prime to x ?” and “How many positive proper fraction with a denominator of x; ( x 2) are reduced to lowest terms?” are the same.

The Sum of Relatively Prime Integers

A new question that has come up is “What is the sum of the positive integers that are less than x and relatively prime to x ”? Using the information from section 2, let N = be the number of positive integers less than x that are relatively prime to x . Then, the sum of the positive integers that are less than x and relatively prime to x is .

Example:

What is the sum of the positive integers that are less than 12 and relatively prime to 12?

Since 12 = 2 2 · 3, we know N = [(2 - 1) · 2 1 ] · [(3 - 1) · 3 0 ] = [1 · 2] · [2 · 1] = 4. Therefore, the sum of the positve integers that are less than 12 and relatively prime to 12 is

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