10 digits

Note that, if the last digit of the product of two number is 1 the pair of digit must be $\large(1,1),(3,7)$ or $\large(9,9)$

For a 10 -digit number , we have 4 option , $\{1,3,7,9\}$ for the $1_{st}$ 9 place.

If the 10-digit positive integer is $\overline{abcdefghij}$ then ,we have 4 option for each of these digit - $a,b,c,d,e,f,g,h,i$ , but we have to put a fix number for $j$ . So, there are $4\times4\times4\times4\times4\times4\times4\times4\times=4^9=\boxed{262144}$ such a beautiful number.

Bonus: There are $4^{n-1}$ , n -digit beautiful number.