Problems
Let \(n\) be any whole number. Prove that the product \((n+1)(n+2)\cdots(2n)\) is divisible by \(2^n\).
A battle of the captains was held at a maths battle. The task was to write the smallest number that is divisible by \(60\) and it’s digits are only ’\(1\)’s or ’\(0\)’s. What is the answer?
Robinson Crusoe and Friday are playing cards. Friday takes \(9\) cards numbered \(1\) to \(9\) and shuffles them. Then he lays them out in a row, making a \(9\)-digit number. Robinson notices something surprising: this number is divisible by \(9\). Was this a coincidence, or will it always be divisible by \(9\)?
At Willy Wonka’s chocolate factory, sweets are always packed into boxes of \(3\).
One day, Charlie, Veruca and Augustus each bring a bag of sweets to the factory. Charlie has some number of sweets, Veruca has one more sweet than Charlie, and Augustus has one more than Veruca.
When they put all their sweets together, will they be able to pack them perfectly into boxes of \(3\), with no sweets left over?
Friday shows Robinson Crusoe a magic trick:
He asks Robinson to write down any 15 whole numbers of his choice on a piece of paper. Then Friday looks at the list, and is always able to pick two of the numbers so that, when one is subtracted from the other, the result is a multiple of \(13\).
Can you explain why this trick works?
Alice was playing on the \(5\times 5\) lights out board and obtained this light pattern:
how did she obtain it?
After some playing with the \(3\times 3\) board, Sam guessed that there were \(900\) different light patterns that could be obtained by playing on this board. Was he right?
Suppose that \(x_1+y_1\sqrt{d}\) and \(x_2+y_2\sqrt{d}\) give solutions to Pell’s equation \(x^2-dy^2=1\) and \(x_1,x_2,y_1,y_2\geq 0\). Show that the following are equivalent:
\(x_1+y_1\sqrt{d} < x_2+y_2\sqrt{d}\),
\(x_1<x_2\) and \(y_1<y_2\),
\(x_1<x_2\) or \(y_1<y_2\).
If Pell’s equation \(x^2-dy^2 = 1\) has a nontrivial solution \((x_1,y_1)\), show that it has infinitely many distinct solutions.
Show that there are infinitely many triples of consecutive integers, each of which is a sum of the square of two integers.