I have written \(5\) composite (not prime and not \(1\)) 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 \(100\).
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 \(d = \mathrm{GCD} (a, b)\).
Prove that if \((p, q) = 1\) and \(p/q\) is a rational root of the polynomial \(P (x) = a_nx^n + \dots + a_1x + a_0\) with integer coefficients, then
a) \(a_0\) is divisible by \(p\);
b) \(a_n\) is divisible by \(q\).
There is a very, very fast way of computing the greatest common divisor of two positive integers. It was in fact known even to the Greeks two thousand years ago. This procedure is called the Euclidean algorithm, named after Euclid, a famous ancient Greek mathematician.
The algorithm works as follows. Take two positive integers \(a,b\). Let’s say \(a\geq b\).
Calculate the remainder of \(a\) when divided by \(b\). Call it \(r_1\).
Calculate the remainder of \(b\) when divided by \(r_1\). Call it \(r_2\).
Calculate the remainder of \(r_1\) when divided by \(r_2\). Call it \(r_3\).
Continue to divide the remainder from two steps prior by the remainder from the last step, until...
The remainder \(r_n\) is divisible by \(r_{n+1}\). The Euclidean algorithm stops now and \(r_{n+1}\) is \(\gcd(a,b)\).
Show that there is indeed some natural number \(n\) such that \(r_n\) is divisible by \(r_{n+1}\), so that the Euclidean algorithm must stop eventually. Furthermore, show that \(r_{n+1}\) is actually \(\gcd(a,b)\) (otherwise it is all in vain!).
Suppose that \(p\) is a prime number. How many numbers are there less than \(p^2\) that are relatively prime to \(p^2\)?