Suppose that \(p\) is a prime number.
a) How many numbers that are less than \(p\) are relatively prime to it?
b) How many numbers that are less than \(p^2\) are relatively prime to it?
I have written \(5\) non-prime numbers on a piece of paper and hidden it in a safe locker. Every pair of these numbers is relatively prime. Show that at least one of these numbers has to be larger than a \(100\).
Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.
Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).