Prove that the number of the sequence 121,12321,1234321,...... are each a perfect square of odd integer.

Prove that the sum of two odd numbers is even.

Find the number of divisors of the number N=2^3 .3^5 .5^7 .7^9 which are perfect squares.

When a group of soldiers are arranged in 4 ddep hollow squares the number of soldiers in the front row is 16 more than the number of soldiers in the front row when they are arranged in a perfect square. Find the number of soldiers.

The conjugate irrational number of a + sqrtb , where b is not a perfect square, is

Prove that when 1 is added to the sum of n terms of the series {8+16+24+…}, the result will be perfect square.

Find the sum of the integers between 90 and 890 which are perfect squares.

Find the sum of the integers between 90 and 890 which are perfect squares.

The number which is added with x^2 + x to make addition a perfect square is

Let f(x) = Ax^2+Bx+C where A,B,C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the number 2A, A+B and C are all integers. Conversely, prove that if the numbers 2A, A+B and C are all integer then f(x) is an integer whenever x is an integer.