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How many base-10 integers are exactly 4 digits in their base-3 representation and exactly 2 digits in their base-6 representation?

May 21, 2019

#1
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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.

May 21, 2019

#1
+4

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.

Melody May 21, 2019