Repeatedly divide the given number by two, rounding down, and add up the results. The reasoning is as follows:
Of the numbers {1,...,19}, [19/2] = 9 are even and therefore contribute a 2 to the factorization of 19!.
Of those, [9/2] = 4 are multiples of 4 and therefore contribute another factor of 2.
Of those, [4/2] = 2 are multiples of 8 and contribute another factor of 2.
Of those, [2/2] = 1 is a multiple of 16 and contributes another factor of 2.
So 2^(9+4+2+1) = 2^16 is the largest power of 2 that divides 19!.

A related puzzle: Show that for all natural numbers N,
N = (the largest k for which 2^k divides N!) + (the number of ones in the binary representation of N). E.g. 19 = 16 + 3.

Repeatedly divide the given number by two, rounding down, and add up the results. The reasoning is as follows:

Of the numbers {1,...,19}, [19/2] = 9 are even and therefore contribute a 2 to the factorization of 19!.

Of those, [9/2] = 4 are multiples of 4 and therefore contribute another factor of 2.

Of those, [4/2] = 2 are multiples of 8 and contribute another factor of 2.

Of those, [2/2] = 1 is a multiple of 16 and contributes another factor of 2.

So 2^(9+4+2+1) = 2^16 is the largest power of 2 that divides 19!.

A related puzzle: Show that for all natural numbers N,

N = (the largest k for which 2^k divides N!) + (the number of ones in the binary representation of N). E.g. 19 = 16 + 3.

Or more generally:

N = (b - 1) * (the largest k for which b^k divides N!) + (the sum of the digits in the base b representation of N)

And still another related problem,

Consider the generation of n rows of Pascals triangle and the coefficients of (a+b)^n as n goes from 0 to ...

i.e rows

1 0

1 1 1

1 2 1 2

1 3 3 1 3

the number of odd coefficients is equal to

2^(sum of 1's in binary representation of n)

Example. n=3 = 11 base 2. sum of 1's =2

2 squared =4 odd coefficients in row 3.

row 128 has 2 odd numbers (the 1's).

These last items thanks to French Mathematician Adrien Legendre (1752-1833)

Anyone for proofs of same?

Sorry for the messed up triangle. The right most terms were in a column labeled Rows.