Calculate \((3141,1592)\) and \((10001,100083)\).

Find two different solutions of \(299x+247y=13\).

Prove: If \(c|ab\) and \((c,a)=d\), then \(c|db\).

Student A says, “I’ve been looking for a half hour for \(n\) such that \(n\) and \(n+20\) have a greatest common divisor of \(7\) and I haven’t found one. I think I’ll program it for the computer.” Student B says, “The computer won’t find one, either.” How did B know that?

Let \((a,b,c)\) denote the greatest common divisor of \(a\), \(b\), and \(c\).

1. Prove that \((a,b,c)=((a,b),c)\).
2. If \((a,b) = (b,c) = (a,c)=1\), prove that \((a,b,c)=1\).
3. Show that the converse of (b) is false.

A man came into a post office and said to the clerk, “Give me some \(13\)-cent stamps, one-fourth as many \(9\)-cent stamps, and enough \(3\)-cent stamps so this \(\$5\) will pay for them all.” How many stamps of each kind did he buy?

Find and prove a theorem inspired from the following facts: \[\begin{align} 3^2 + 4^2 &= 5^2, \\ 10^2 + 11^2 + 12^2 &= 13^2 + 14^2, \\ 21^2 + 22^2 + 23^2 + 24^4 &= 25^2 + 26^2 + 27^2, \\ 36^2 + 37^2 + 38^2 + 39^2 + 40^2 &= 41^2 + 42^2 + 43^2 + 44^2. \\ \end{align}\]