Divisibility

Find all positive integers n such that a linear sequence divides a quadratic polynomial. Try using modular arithmetic to rewrite the statement to factor. W. Sierpinski's 250 Problems - problem # 1

Show there are infinite positive integers X for which a quadratic is divisible by five and thirteen. Solve by Chinese Remainder Theorem.

Show there are infinite positive integers n for which a quadratic is divisible by five and thirteen. Solve by system of equations.

Show there are infinite positive integers x for which a quadratic is divisible by three and five. Solve by system of equations. Sources: blackpenredpen - YT Channel

Show by induction there are infinite positive integers n for which a quadratic is divisible by five and thirteen. From W. Sierpinski's 250 problems, 1970, problem # 3.

Does a product of primes divide an expression? We justified our steps with a discussion of Euler's phi function and Fermat's little theorem. To clarify, if we consider all of the integers, and then if we perform mod 5, then all of those integers are congruent to 0, 1, 2, 3, 4 (mod5). Which applies to (0, 1, 2...n-1) (modn). Sources: Socratica - YT Channel W. Sierpinski - Problems in Number Theory, Polish Academy of Sciences, 1970; problem 7