IIT JEE , AIEEE Forum - Mathematics , Algebra

mukundmadhav

See Base number systems are basically number systems which use only a particular number of numerals
Like decimal is base 10, it uses 0,1,2...9
Binary is base 2. It uses 0 and 1
Similarly base 6 will use 0,1,2,3,4,5
For a three digit number in base 6, the number of numbers is 216(Because there are six choices each for each of the three digits)
But the first number is 000, so the largest three digit number in base 6 is 215(which is represented in base 6 as 555). Now You need a three digit number which is a perfect square. Greatest number which is a square and less than 215 is 196, which is 14^2.

Now if you want 196 to be expressed as a two digit number, you'll need a number system which has so many numerals that it is possible. Let the number sytem be k. Number of two digit numbers in this system will be k^2. Now 196 should be strictly less than k^2
So k=15..
Now to express 15 in base 7 you proceed like this
0, 1, 2, 3, 4, 5, 6
And instead of 7, you now write 10
So 7 in base 10 = 10 in base 7 :P :P :P
Going on, 10, 11, 12, 13, 14, 15, 16
Now since you can't write 17(Because 7 does not exist in base 7)
you write 20
Hence the answer is 20

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