# Thread: Find smallest n for which this is possible?

1. ## Find smallest n for which this is possible?

An $\displaystyle n$ digit number is a positive number with exactly $\displaystyle n$ digits. At least nine hundred $\displaystyle n-digit$ numbers are to be formed using only the three digits 2, 5 and 7. Find the smallest $\displaystyle n$ for which this is possible.

2. How many one digit numbers use only 2,5,7?

3, as you have 3 choices for the digit.

How many two digit numbers use only 2,5,7?

3*3, as you have 3 choices for the first digit, and 3 choices for the second.
(22, 25, 27, 52, 55, 57, 72, 75, 77)

Similarly, there are 3*3*3 three digit numbers that use only 2,5,7.

So..

3. Hello, fardeen_gen!

This one is easier than you think . . .

An $\displaystyle n$-digit number is a positive number with exactly $\displaystyle n$ digits.
At least nine hundred $\displaystyle n$-digit numbers are to be formed using only the three digits 2, 5 and 7.
Find the smallest $\displaystyle n$ for which this is possible.
We can baby-talk our way through it . . .

There are $\displaystyle 3^1 = 3$ one-digit number that can be formed.
There are $\displaystyle 3^2 = 9$ two-digit numbers that can be formed.
There are $\displaystyle 3^3 = 27$ three-digit numbers that can be formed.
There are $\displaystyle 3^4 = 81$ four-digit numbers that can be formed.
There are $\displaystyle 3^5 = 243$ five-digit numbers than can be formed.
There are $\displaystyle 3^6 = 729$ six-digit numbers than can be formed.
There are $\displaystyle 3^7 = 2187$ seven-digit numbers than can be formed.

Therefore: .$\displaystyle n = 7$

I'll let someone else show you a more sophisticated method.