Let ^@S = \{1, 2, 3, ....., 46, 47\}^@. What is the maximum valu | Math Question and Answer

Answer:

$18$

Step by Step Explanation:

Given $S = \{1, 2, 3, ....., 46, 47\}$ .
Now, we know that the product of any two $2$ -digit numbers is either equal to or more than $100$ .
If $n$ numbers are chosen from $S$ and arranged as per the question, no two $2$ -digit numbers are adjacent.
Therefore, the two numbers adjacent to a $2$ -digit number must be single-digit numbers.
A maximum of nine $1$ -digit numbers can be chosen from $S$ and a $2$ -digit number can fit in between any two $1$ -digit numbers. There will be $9$ such places between any two $1$ -digit numbers.
Without loss of generality, let us place the numbers $1, 2, 3,...,9$ in the ascending order. Now place the numbers $10, 11, 12, ...,18$ in between these numbers such that $18$ is placed between $1$ and $2, 17$ is placed between $2$ and $3, 16$ is placed between $3$ and $4$ and so on to ensure that the product of any two adjacent numbers is less than $100$ .
Therefore, one can choose a maximum of $18$ numbers $($ nine $1$ -digit numbers and nine $2$ -digit numbers $)$ from $S$ and arrange them in a circle in such a way that the product of any two adjacent numbers in the circle is less than $100$ .
Hence, the maximum value of $n$ is $18$ .

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