Here is my number today:

From Random Graphic Images

See this post to understand why I think that figuring out why the prime factorization of my Char-Grill number is cool.

Also, since this was not too hard to figure out. I started thinking what else can I do with this number.

A good idea is to compute the value of Euler’s Totient Function at that number.

Here is the definition of \varphi(n)

The number of positive integers less than or equal to n that are co-prime to n.

Two positive integers are co-prime if they share no common factors other than one.

So it turns out that \varphi(n) is multiplicative which means \varphi(mn) = \varphi(m)\varphi(n)

And it also turns out that:

\varphi(p^k) = p^k(1-1/p) = p^{k-1}(p-1)

If you put this together with k=1 and note that

1-1 = 0  and x^{0}=1

then \varphi(p) = p-1

And applying the totient function to the prime factorization of a number yields:

\displaystyle\varphi(n) = \displaystyle\varphi(\displaystyle\prod_{i=1}^m(p_i^{k_i})) where m is the number of prime factors of n

And now since 1781 = 43^1 \times 167^1 we get

\varphi(1781) = \varphi(43) \times \varphi(167) = (43-1)(167-1) = (42)(168) = 7056