Number Theory

Primes, divisibility, and modular arithmetic.

  1. Let Sᵢ denote the set of all integers n such that 100i ≤ n < 100(i+1). For example S₄ contains 400,401, 402, 403,...,499. How many such sets from S₀ to S₉₉₉ do not contain a perfect square.
  2. How many positive perfect squares less than 10⁶ are multiples of 24?. (A perfect square is an integer that is the square of an integer)
  3. Determine the number of positive integral values of p for which there exists a triangle with sides a, b and c satisfying a²+(p²+9)b²+9c²-6ab-6pcb=0.