RMO number theory class 1

 Modulo of perfect square numbers.


Is the converse true?
No.
For e.g. 20 mod 4 is 0 but 20 is not a perfect square.


So we proved this by taking mod 8 which is 5 hence not a P.S.
mod 9 gives 7 => not a P.C.

How do you compute mod 9?
Simply take some of digits and compute mod.
Because abc = 100a + 10b + c.





Q3

solution:





Answer: 59


Q4

solution:
partial solution:

So it is divisible by 3 if n is multiple of 3.
But if n is odd it's not div by 11(alternate sum of digits not  0).
So n is even.
So the number is 9..{6k 2s}...9
Let's express it as GP and compute the sum.
Let the sum be N.
Once we do that we will get 9N = 61 + 830 *10^n.
If 9N is div by 61, N will be also.
9N mod 61 = 0 + 37*10^n mod 61
37 and 61 are co prime so 10^n has to be div by 61 but that's not possible.
So no such 'n' exists.

Q5


solution


Magic square(sum of all rows,columns,diagonals same)
Any magic square made with digits 0 to 9 holds this property:








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