Show that numbers \(12n+1\) and \(12n+7\) are relatively prime.
If natural numbers \(a,b\) and \(c\) are lengths of the sides of a right triangle (such that \(a^2+b^2=c^2\)), show that at least one of these numbers is divisible by \(3\).
Tom got a really bad grade from the last test and once he got the test back, he started to tear it up. He is tearing it into little pieces in the following manner: He picks up a piece and tears it into either \(4\) or \(10\) smaller pieces. Can he eventually have exactly 200,000 pieces?
Show that any natural number has the same remainder when divided by \(3\) as the sum of its digits.
Show that \(n^3 + 2n\) is divisible by \(3\) for a natural \(n\).
Prove that if \(a^3- b^3\), for \(a\) and \(b\) natural, is divisible by \(3\), then it is divisible by \(9\).
What time is it going to be in \(2019\) hours from now?
What is a remainder of \(1203 \times 1203 - 1202 \times 1205\) when divided by \(12\)?
Show that a perfect square can only have remainders 0 or 1 when divided by 4.
Convert 2000 seconds into minutes and seconds.