A number with sixteen digits

We will show that every sixteen-digit number is a beautiful number. Let $a_1a_2a_3a_4\dots a_{15}a_{16}$ be a sixteen-digit number. If it has a digit which is a square number, then it is a beautiful number. If it has only digits from ${2, 3, 5, 6, 7, 8}$ , then we can produce the $a_1, a_1*a_2, a_1*a_2*a_3, \dots a_1*a_2*a_3*\dots a_{16}$ numbers in a $2^v*3^x*5^y*7^z$ formula.

If $a_1*a_2*\dots a_n$ 's and $a_1*a_2*\dots a_k$ 's $(n>k)$ $v, x, y, z$ 's parity is pairwise equal (parity means even or odd), then $\frac{a_1*a_2*\dots a_n}{a_1*a_2*\dots a_k}$ is a square number, and it can be made by $a_k*a_{k+1}*\dots*a_n$ .

The $v-x-y-z$ 's parity has sixteen options: (E:even, O:odd)

EEEE, EEEO, EEOE, EOEE, OEEE, EEOO, EOEO, OEEO, EOOE, OEOE, OOEE, EOOO, OEOO, OOEO, OOOE, OOOO.

But EEEE is not possible because it is a square number. So there has to be two of the $a_1, a_1*a_2, a_1*a_2*a_3, \dots a_1*a_2*a_3*\dots a_{16}$ which parities are pairwise equal.

So each sixteen-digit number is beautiful and there are $9000000000000000$ $(9*10^{16})$ sixteen-digit numbers.