Problem 26

Prove that every composite number has a proper factor less than or equal to its square root.

Problem 25

Show that 43ⁿ + 83 x 92^3n-1 is divisible by 7 for all positive integers n.

Problem 21

Prove that the a composite n non divisible by   is a semiprime.

Problem 20

Find the last digit of 43⁴³.

Problem 19

Find the last digit of 2⁹⁹⁹.

Problem 14

Fill in the missing digit to make the number 66* divisible by 17.

Problem 13

Prove that the square root of 201720172017  is an irrational number.

Problem 9

Fill in the missing digit to make the number 3*55 divisible by 13.

Problem 8

Fill in the missing digit to make the number 865* divisible by 7.

Problem 7

Fill in the missing digits to make the number 766776** divisible by 11.

Problem 5

A positive integer n > 1 can be represented a product of prime powers:

where p₁ < p₂ < ... < p_k are primes and the α_i are positive integers. Find how many factors are in n.