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1 ’s are given 100 times, 2 ’s are given 100 times and 3’s are given 100 times. Now numbers are made by arranging these 300 digits in all possible ways. How many of these numbers will be perfect squares?

  • a)
    10101

  • b)
    1001001

  • c)
    100010001

  • d)
    None of these

Correct answer is option 'D'. Can you explain this answer?
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1 ’s are given 100 times, 2 ’s are given 100 times and 3&r...
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  1. Any number formed by this method is clearly divisible by 3.

  2. Since it needs to be a square, it should be divisible by (3)[2*k]. k varies over the natural numbers.

  3. Now consider the original number. It has hundred 1’s, hundred 2’s and hundred 3’s. Sum of these digits is 600. This is not divisible by 9. Hence number is not divisible by 9.

  4. If a number is divisible by (3)[2*k], it is divisible by 3k.

  5. This number is not divisible by 3k for any k > 1. 



Hence it is not a perfect square for any arrangement.
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1 ’s are given 100 times, 2 ’s are given 100 times and 3&r...
What if it is divisible by perfect square?? How do we know how many perfect squares will be there??
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1 ’s are given 100 times, 2 ’s are given 100 times and 3&r...
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