Find the one-hundredth-smallest positive integer that can be written using only the digits $1$, $3$, and $5$ in base $7$.
So, I came across this problem in an old contest I did in 2018, and I still don't know how to solve it. I honestly don't even know where to start. Could someone please help me? Thank you in advance !
Best Answer
You can just list them out in base $7$ first, and I think this won’t take you too much time. In fact, it’s actually fast.
1-digit numbers in base $7$ : There are $3$.
2-digit numbers in base $7$ : There are $9$.
3-digit numbers in base $7$ : There are $27$.
And now consider 4-digit numbers in base $7$.
Starts with $1$ (which is $1$ _ _ _ ) : There are $27$.
Starts with $3$ (which is $3$ _ _ _ ) : There are $27$.
So far we have $93$ numbers satisfying the condition.
Start with $5$ : $5111,5113,5115,5131,5133,5135,5151$
The answer is $5151$(base $7$) , which is $1800$ in base $10$.