Arithmetic and Number Theory

Find the hightest power of 2 that divides 21! (factorial).

Does the uniqueness of prime factorization hold in other number sets?

How many positive integers less than 1000 are divisible neither by 5 nor by 7?

Is it possible to find five positive whole numbers 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 such that there is no subset of them with a sum divisible by 5?

Show that 15𝑥² − 7𝑦² = 9 has no integer solution.

Can you arrange all the numbers from 1 to 17 so that the sum of adjacent numbers is always a square number?

Can a number 𝑁, consisting of 600 sixes and some zeros, be a square number?

Can you represent the number one billion as the product of two integers m and n such that neither m nor n ends in the digit 0?

Without a calculator, smartphone, or computer, can you determine if the number 3, 141, 592, 653, 589, 793 (that is, 3 quadrillion 141 trillion 592 billion 653 million 589 thousand 793) is a square number?