Let S={1,2,3,.....,46,47}S={1,2,3,.....,46,47}S={1,2,3,.....,46,47}. What is the maximum value of nnn such that it is possible to select nnn numbers from SSS 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?100?100?


Answer:

181818

Step by Step Explanation:
  1. Given S={1,2,3,.....,46,47}S={1,2,3,.....,46,47}S={1,2,3,.....,46,47}.
    Now, we know that the product of any two 222-digit numbers is either equal to or more than 100100100.
  2. If nnn numbers are chosen from SSS and arranged as per the question, no two 222-digit numbers are adjacent.
    Therefore, the two numbers adjacent to a 222-digit number must be single-digit numbers.
  3. A maximum of nine 111-digit numbers can be chosen from SSS and a 222-digit number can fit in between any two 111-digit numbers. There will be 999 such places between any two 111-digit numbers.
    Without loss of generality, let us place the numbers 1,2,3,...,91,2,3,...,91,2,3,...,9 in the ascending order. Now place the numbers 10,11,12,...,1810,11,12,...,1810,11,12,...,18 in between these numbers such that 181818 is placed between 111 and 2,172,172,17 is placed between 222 and 3,163,163,16 is placed between 333 and 444 and so on to ensure that the product of any two adjacent numbers is less than 100100100.
  4. Therefore, one can choose a maximum of 181818 numbers (((nine 111-digit numbers and nine 222-digit numbers))) from SSS and arrange them in a circle in such a way that the product of any two adjacent numbers in the circle is less than 100100100.
  5. Hence, the maximum value of nnn is 181818.

You can reuse this answer
Creative Commons License