The product of two positive numbers \(a\) and \(b\) is greater than \(100\). Prove that at least one of the numbers is greater than \(10\).
Write the contrapositive of the statement “If it is sunny outside, then I put on sunscreen and wear sunglasses”
What is the contrapositive of the statement: “If the temperature is above \(40^\circ\)C or below \(-10^\circ\)C, then it is not safe to go outside."
Some lines are drawn on a large sheet of paper so that all of them meet at one point. Show that if there are at least \(10\) lines, then there must be two lines whose angle between them is at most \(18^\circ\).
A whole number \(n\) has the property that when you multiply it by \(3\) and then add \(2\), the result is odd. Use proof by contrapositive to show that \(n\) itself must be odd.
Show that the sum of any \(100\) consecutive numbers is a multiple of \(50\) but not a multiple of \(100\).
Alice sums \(n\) consecutive numbers, not necessarily starting from \(1\), where \(n\) is a multiple of four. An example of such a sum is \(5+6+7+8\). Can this sum ever be odd?