Upon the installation of a keypad lock, each of the 26 letters located on the lock’s keypad is assigned an arbitrary natural number known only to the owner of the lock. Different letters do not necessarily have different numbers assigned to them. After a combination of different letters, where each letter is typed once at most, is entered into the lock a summation is carried out of the corresponding numbers to the letters typed in. The lock opens only if the result of the summation is divisible by 26. Prove that for any set of numbers assigned to the 26 letters, there exists a combination that will open the lock.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.
What time is it going to be in \(2025\) hours from now?
Prove that the product of five consecutive integers is divisible by \(30\).
Prove that if \(n\) is a composite number, then \(n\) is divisible by some natural number \(x\) such that \(1 < x\leq \sqrt{n}\).
The natural numbers \(a,b,c,d\) are such that \(ab=cd\). Prove that the number \(a^{2025} + b^{2025} + c^{2025} + d^{2025}\) is composite.
Prove that for an arbitrary odd \(n = 2m - 1\) the sum \(S = 1^n + 2^n + ... + n^n\) is divisible by \(1 + 2 + ... + n = nm\).
Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by \(24\).
Mattia is thinking of a big positive integer. He tells you what this number to the power of \(4\) is. Unfortunately it’s so large that you tune out, and only hear that the final digit is \(4\). How do you know that he’s lying?
You might want to know what day of the week your birthday is this
year. Mathematician John Conway invented an algorithm called the
‘Doomsday Rule’ to determine which day of the week a particular date
falls on. It works by finding the ‘anchor day’ for the year that you’re
working in. For \(2025\), the anchor
day is Friday. Certain days in the calendar always fall on the anchor
day. Some memorable ones are the following:
‘\(0\)’ of March - which is \(29\)th February in a leap year, and \(28\)th February otherwise.
\(4\)th April, \(6\)th June, \(8\)th August, \(10\)th October and \(12\)th December. These are easier to remember as \(4/4\), \(6/6\), \(8/8\), \(10/10\) and \(12/12\).
\(9\)th May, \(11\)th July, \(5\)th September and \(7\)th November. These are easier to see as
\(9/5\), \(11/7\), \(5/9\) and \(7/11\). A mnemonic for them is “9-5 at the
7-11".
Then find the nearest one of these dates to the date that you’re looking
for and find remainders.
For example, \(\pi\) day, (\(14\)th March, which is written \(3/14\) in American date notation. It’s also Albert Einstein’s birthday) is exactly \(14\) days after ‘\(0\)’th March, so is the same day of the week - Friday in \(2025\).
What day of the week will \(25\)th December be in \(2025\)?