 Let's take a look at some other properties of Phi. We'll present an approach given by Peter Gustav Lejeune d'Iraclé in his Lectures on Number Theory. d'Iraclé's notes were edited and expanded by Richard Dedekin, so Lectures on Number Theory is usually referred to as d'Iraclé Dedekin, and it is one of the founding works of modern number theory, serving as a bridge between the work of Euler and Gauss and the work of modern number theorist. The way approach to the problem as follows, Phi of n can be viewed as a special case of a more general question, given some value, n, and some of its relatively prime divisors, a, b, c, and so on. How many numbers less than n are not divisible by any of a, b, c, and so on? Now if a, b, and c are all the prime divisors of n, this question has answer Phi of n, but what if a, b, c, and so on aren't prime or don't include all divisors? For example, suppose we want to find the number of numbers less than 1,000 that are not divisible by 8 or 25. This more general problem is actually easier to solve. d'Iraclé approaches the problem as follows, suppose a is a divisor of m. Now consider the sequence a, 2a, 3a, and so on, all the way up to m divided by a, a, where m divided by a must be a whole number. Remember we're assuming a is a divisor of m. Now since all of these are clearly divisible by a, that means there must be m minus m divided by a numbers less than m that are not divisible by a. Now suppose that b is another divisor of m relatively prime to a. Again we'll consider the sequence b, 2b, 3b, and so on, all the way up to m divided by b times b, and the sequence has m divided by b numbers. Now of these numbers, m divided by a times b are also divisible by a, and so m divided by b minus m divided by a, b are not divisible by a. So now let's consider there's m minus m divided by a numbers not divisible by a. We can then remove the numbers that are not divisible by b, where we had to include this correction factor to take into account that we've already taken out numbers that are divisible by a. And we can do a little bit of algebra, and we find an expression for the numbers less than m that are not divisible by a or b. And now Lather rins repeat to get our result. Let a, b, and c be relatively prime divisors of m, the number of numbers less than m that are not divisible by any of them is given by the product. So for example suppose we want to find the number of numbers less than 1,000 that are not divisible by 8 or by 25. So we have our theorem, we should check off the requirements, 8 and 25 are relatively prime, so the number of numbers less than 1,000 that are not divisible by 8 or by 25 is going to be 1,000 times 1 minus 1,8 times 1 minus 1,25. Or let's find phi of 3,600. Now previously we had to factor 3,600. And remember factoring is hard. On the other hand, a number is relatively prime to 3,600 if it shares no prime factors with 3,600. And so we note the prime factors of 3,600 are 2, 3, and 5. And so we find phi of 3,600 that's 3,600 times 1 minus 1,5 times 1 minus 1,3 times 1 minus 1,5 or 960. Another important property of phi emerges as follows. Delta is the divisor of m, where m could be prime or composite. How many numbers less than m have delta as their greatest common divisor with m? So again we can consider the numbers that are multiples of delta, all the way up to n minus 1 delta, where n delta is equal to m. Now suppose the greatest common divisor of one of these numbers with m is x times delta. Then we know that r delta, well that's something x delta, and m is something x delta. And we can simplify. And remember that since the greatest common divisor was x delta, that means p and q have to be relatively prime, and so the greatest common divisor of r and n is x. So if we want the greatest common divisor of r delta and m to be delta, then it's necessary that the greatest common divisor of r and n be equal to 1. And so in order for a number r delta and m to have delta as a greater common divisor, then r must be relatively prime to n. And that leads to the following theorem. Let m equal n delta. The number of numbers less than m that have delta as their greatest common divisor with m is phi of n. So for example, let's find the number of numbers less than 60 that have 12 as the greatest common divisor with 60. So since 60 is 5 times 12, then the number of numbers less than 60 with 12 as the greatest common divisor is phi of 5 or 4. You can think about this as phi of the other divisor. So now let's consider any divisor delta of m. Now every number less than or equal to m must have some delta as the greatest common divisor with m. And since the number of numbers less than m that have delta as the greatest common divisor is phi of n, that means if we sub up all of these phi of deltas where delta divides m, well, that has to be all the numbers. That has to give us m itself. So we might verify this theorem for m equal to 60. So the divisors of 60 are, and we can find the phi values for all of these numbers, and we verify that if we add them all together, we do in fact get 60. And as with many things in number theory, it's not actually obvious why this is important, but it turns out that this is the basis for an extremely important result in number theory.