How many base-10 integers are exactly 4 digits in their base-3 representation and exactly 2 digits in their base-6 representation?
How many base-10 integers are exactly 4 digits in their base-3 representation and exactly 2 digits in their base-6 representation?
4 digits in base 3
1000 base 3 = 3^3 = 27 base 10
2222 base 3 = 2*27+2*9+2*3+2 = 80
So it has to be between 27 and 80 base 10
and exactly 2 digits in their base-6
10base 6 = 6 base 10
55base 6 = 6^2-1 = 35 base 10
So it has to be between 6 and 35 inclusive base 10
So to meet both these requirements the number must be between 27 and 35 base 10 inclusive.
So 9 numbers qualify.
How many base-10 integers are exactly 4 digits in their base-3 representation and exactly 2 digits in their base-6 representation?
4 digits in base 3
1000 base 3 = 3^3 = 27 base 10
2222 base 3 = 2*27+2*9+2*3+2 = 80
So it has to be between 27 and 80 base 10
and exactly 2 digits in their base-6
10base 6 = 6 base 10
55base 6 = 6^2-1 = 35 base 10
So it has to be between 6 and 35 inclusive base 10
So to meet both these requirements the number must be between 27 and 35 base 10 inclusive.
So 9 numbers qualify.