Suppose that p and q are prime numbers and that n=p q . Use the principle of inclusion-exclusion

step 1 Let P and Q be prime numbers and let N be the product of P and Q. How many positive integers les

Suppose that p and q are prime numbers and that n=p q . Use...

Key Concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a combinatorial method used to count the number of elements in the union of overlapping sets. It works by adding together the sizes of individual sets and then subtracting the sizes of all pairwise intersections (and further, adding or subtracting intersections of three or more sets, if necessary) to avoid overcounting. This principle is especially useful when counting elements that satisfy at least one of several properties, as it systematically corrects for multiple counting of elements that fulfill more than one property.

Relatively Prime Numbers

Relatively prime numbers are pairs (or groups) of integers that share no common factors other than 1. In other words, the greatest common divisor (gcd) of two numbers is 1. This concept is central in many areas of number theory, including the study of Euler's Totient Function, which counts how many numbers up to a given limit are relatively prime to a particular number.

Euler's Totient Function

Euler's Totient Function, often denoted as ?(n), counts the number of positive integers up to n that are relatively prime to n. It is an important arithmetic function in number theory with applications in topics such as modular arithmetic and cryptography. When n is the product of two distinct prime numbers, the function simplifies due to the multiplicative nature of the totient function, and the count can be derived using a formula based on subtraction of numbers divisible by these primes.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime components. Since primes are the basic building blocks of the integers, understanding the prime factors of a number is crucial for various aspects of number theory, including determining the set of numbers relatively prime to n. This concept is fundamental in proofs and applications involving Euler's Totient Function and the use of the Inclusion-Exclusion Principle to count numbers with specific divisibility properties.

Transcript

00:01 Let p and q be prime numbers and let n be the product of p and q.

00:05 How many positive integers less than n are relatively prime to n? well, let's give it a try with some numbers, and then maybe we'll be able to figure it out.

00:17 Let's let p equals 7 and, oops, and q equal, let's say, 19.

00:34 Is 7 times 19, which is 70 and 63, so 133.

00:44 So there are 132 positive integers, less than in, which are relatively prime.

01:06 Okay, now relatively prime, in case you don't know, means that the two numbers have nothing in common.

01:12 So like 100, no factors in common.

01:17 So like 100 and 133 are relatively prime, even though 100 is not prime, but it has nothing, no factors in common with 133.

01:29 Okay, well, since 133 is 7 times 19, we know that no multiples of 7 will be relatively prime.

01:37 So here are the multiples of seven that are less than 133.

01:47 So 7, 14, 21, 28, 35, 42, 49, 59, 56, 63, 70, 77, 84, 91, 98, 105, 105, 10, 11, 112, 112, 119, 126, and then the next one will be 133.

02:21 So there's 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 17.

02:30 Oh, oh yeah, 17, 18.

02:35 So there's 18 of them.

02:39 Okay, now i could have figured that out without counting because this is 7 times 1, 7 times 2, 7 times 3.

02:46 I multiply 7 times every number up to 18.

02:50 So none of these are relatively primed to 133 because they share a factor of 7.

02:59 Okay, then also no multiples of 19 are relatively prime.

03:06 So multiples of 19, 38, 57, 76, 95 plus 215, 115, 114, and then the next one will be 133...

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