Question

how many three-digit base-10 numbers also have base-9 and base-11 representations that are also three-digits?

Answer 1

Hi,

608

Explanation:

In base 10: numbers are in range(100,999) => 999-100+1=900 numbers

In base 9: numbers are in range(100,888)

=>8*9^2+8*9^1+8 - 9²+1= 648 numbers

In base 11: numbers are in range(100,AAA)

=> 10*11²+10*11+10-11²+1 =1330-121+1=1210 numbers

three-digit base-10 ,three-digit base-9 and three-digit base-11:

numbers are from range ((100)_11,(888)_9)

=> 728-121+1=608 numbers.