Question

An n digit number is a positive number with exactly n digits. Nine hundred distinct n digit numbers are to be formed using only the three digits 2, 5, and 7. The smallest value of n for which this is possible is
(a) 6
(b) 7
(c) 8
(d) 9

Answer 1

Hint: In this question, first check how many digits are there in the number 900 and then those numbers of digits will become your base of the question through which you need to find the value of n. Once you make the equation, compare it with the given number, and find the value of n.

Complete step-by-step solution:
Here we need to find an n digit number which is a positive number and has exactly n digits. This number can be formed using only 2, 5, and 7, and hence we need to find the smallest value of n.
For $1^{st}$ spot, we have digit 3
For $2^{nd}$ spot, we have digit 3
And for the ${{n}^{th}}$spot, we have digit 3
Hence, the total number of ways to make an n-digit number is ${{3}^{n}}$.
If we look at the question, accordingly, we have
$\begin{align}
  & {{3}^{n}}\ge 900 \\
 &\Rightarrow {{3}^{n}}\ge {{\left( 3 \right)}^{2}}\cdot 100 \\
 &\Rightarrow \dfrac{{{3}^{n}}}{{{3}^{2}}}\ge 100 \\
 &\Rightarrow {{3}^{n-2}}\ge 100 \\
\end{align}$
Now, here we need to take the value of n which satisfies the left-hand side of the equation to the right-hand side.
Hence, we need $n-2\ge 5$
When we solve this, we will get the value of n as greater than or equal to 7.
So, when we take the value of n as 7, we will get the value n – 2 as 5 which makes ${{3}^{5}}=243$ which is greater than 100, and hence it satisfies the equation.
Hence, the smallest value of n is 7.

Note: Here, it is important to figure out the number of digits, which give the base to find the original number. While you are using inequations, carefully write down the conditions to find the value of n. Here we had to choose the value of n such that the value of the base(3) for n to be greater than 100.