KVPY 2016 SA Question 3

$16^5 \times 5^{16} = 2^4 \cdot 2^{16} \times 5^{16} = 10^{16} \times 16$ which has 18 digits.

16^5 . 5^16 =2^20 .5^16 =10^16 .2^4 =32 .10^16 we can easily see here number of digit is 18. Second way is take Log then it will be 5log16+16log5 =20log2 +16log5=6.02 +11.08 then number of digit will be [17.10] +1 =18 Here [ ] is floor funtion.

Since 2 m X 5 m=10...00,where the zeros occur m times(here m=16), and 2 4=16 with 2 digits, there are (2+16) digits, i.e. there are 18 digits.( stands for power)