Perfect Squares  Kamal Lohia

Some Properties (Square of integers)
Unit Digit of a perfect Square cannot be 2, 3, 7 or 8.
Ten's digit of a perfect square is always even except when unit digit of perfect square is 6.
Digital sum of a perfect square is 1, 4, 7 or 9.
All perfect squares are of the form 3a or 3a + 1.
All perfect squares are of the form 4a or 4a + 1.
All perfect squares are of the form 5a or 5a + 1 or 5a  1.
How many numbers of the set S {1, 11, 111, … } are perfect squares?
Only 1. As all other numbers are of the form 4a + 3 which cannot be a perfect square.
How many numbers of the set P{4, 44, 444, 4444, ....} are perfect squares?
again only 4 is perfect square.
How many five digit perfect squares can be formed by using the digits 1, 2, 3, 4, 5 exactly once?
None. As digital sum of the number will be (1 + 2 + 3 + 4 + 5) mod 9 = 6 which is not possible for a perfect square.
Find the number of positive integral pairs (a, b) such that a^{2 }+ b^{2 }= 1000.
As RHS is multiple of 4, both terms at LHS should be of the form 4k i.e. a, b both are multiple of 2.
Also as RHS is of the form 3k + 1, one of a, b is multiple of 3. (Think why??)
As 31^{2 } < 1000 < 32^{2}, none of a, b is more than 31. And also we got to know that one of a, b is multiple of 2 as well as 3 i.e. 6 and other is multiple of 2 only.
Now you can easily check that multiple of 6 can be 6, 12, 18, 24, or 30 only.
18^{2 }+ 26^{2 }= 324 + 676 = 1000, and
30^{2 }+ 10^{2 }= 900 + 100 = 1000
So four cases for (a, b) are possible.
Nimai and Nitai are two brothers who have some mangoes with them to sell. They fix the price of each mango to be equal to the number of total mangoes with both of them together initially. Together they sell all the mangoes and after that they start distributing the money collected in this particular fashion. First Nimai takes a 10 rupee note, then Nitai takes a 10 rupee note and so on. In end it's turn of Nitai who don't get any more 10 rupees. Can you tell me how much rupees he get in his last turn?
As Nimai started the distribution part and took a 10 rupee note first and in the end also, he is able to take a 10 rupee note. That means Total amount, which needs to be a perfect square for n mangoes @ n rupees per mango, is odd multiple of 10 plus some more which is less than 10. That means ten's place digit of the perfect square is ODD. So certainly unit digit of perfect square is 6.