The simplest case is when our number is prime,

in this situation the totient is one less than the number.

This is actually quite obvious, since the fact that it is prime number

means that every positive integer less than it is relatively prime to it.

This result is sufficiently obvious that we can go ahead and

consider this case proven.

So for example, since 89 and 97 are both prime numbers,

the totient of 89 is 88, while the totient of 97 is 96.

What about when N is a product of prime numbers, without any repeats?

In that case, the product of all the prime factors that builds up

in the denominator is equal to N, cancelling it out.

The result is that the totient is the product of factors,

each equal to 1 less than the corresponding prime factor.

The fact that this is correct isn't immediately obvious, but

it's not too hard to reason out.

However the approach we will take in a little bit will make this result

an obvious special case.

Using the two prime numbers from our previous example,

the totient of 8633, which is the product of 89 and

97, is 8448, the product of 88 and 96.

The product of the first ten prime numbers,

which is all prime numbers less than 30, is 6 billion, and

a bunch of change, and its totient is little over 1 billion.

As you can see, our numbers can get very large, very quickly.

In fact the growth of the product of the first k primes

grows at a rate on the order of, but not quite as fast as,

the factorial function, which makes sense if you think about it.