Problems

Prove that the only solution to \(5^a-3^b=2\) with \(a,b\) being positive integers is \(a=b=1\).

Show that Pell’s equation \(x^2-dy^2=1\) has a nontrivial solution.

For the following equations, find the integer solution \((x,y)\) with the smallest possible absolute value of \(y\).

• \(x^2 - 7y^2 = 1\);

• \(x^2 - 7y^2 = 29\).

Find the integer solution \((x,y)\) with the smallest possible absolute value of \(y\). \(x^2 - 2y^2 = 1\);

This equation helps to find all the square-triangular numbers, namely all the numbers that are perfect squares and can be represented as the sum \(1+2+3+...m\) for some \(m\). Finding such a number is equivalent to finding a solution to the equation: \(2n^2 = m(m+1)\). Or finding a solution to the Pell’s equation \(x^2-2y^2 = 1\) for \(x=2m+1\), \(y=2n\).

A rectangle has a perimeter of \(1\). Is it possible that its area is larger than \(1000\)?

Zahra has a \(3\times 3\) grid of little squares. Can she write the numbers \(2,4,6,7,8,10,12,14,16\) inside the little squares - using each number exactly once - so that the sum of the three numbers in every row is the same?

Sam was playing on the \(3\times 3\) lights out board, and starting from all lights off, he pressed every single button on the board. What light pattern did he get in the end?

Which of the triangles has the smallest area?

Find a quiet plan for the \(1\times 5\) “Lights Out" board.

Welcome everybody! In today’s session we will be talking about divisibility tricks. Recall that a number \(a\) is divisible by another number \(b\) if \(a\) divided by \(b\) is a whole number. Often, there are quick ways to check divisibility without doing the full division. For example, a number is divisible by \(5\) if and only if its last digit is \(0\) or \(5\). It is important to remember that this phrase “if and only if" actually means two things:

1. If a number is divisible by \(5\), then its last digit must be \(0\) or \(5\).

2. If a number’s last digit is \(0\) or \(5\), then the number is divisible by \(5\).

It is useful to think about this in terms of there being two directions. In today’s sheet we will see many more such tricks, and remember: usually you will need to prove both directions!