All numbers from $1$ to $150$ (in decimal system) are written in base $6$ notation. How many of these will contain zero's?

Question

Now I know that any number in decimal system which is divisible by the base $6$ will have a $0$ in the unit's place when that number in the decimal system will be converted into base $6$. This gives me a count of total $25$ numbers.
Now I know that there are other numbers in the decimal system which I have left in the previous count of $25$ which when converted into base $6$ notation will have $0$ apart from the unit's place too. How can I count those? What concept I am missing here? Please help me on this !!!!

Thanks in advance !!!

Answer 1

Hints:

• $150_{10} = 410_{6}$.
• How many numbers have a 0 in the units place?
• How many numbers have a 0 in the tens place, and a non-0 in the units place?
• Hence, the total is $ 4\times 6 + 1 + 4\times 5 = 45$.