Observe that \(14\) isn’t a square
number but \(144=12^2\) and \(1444=38^2\) are both square numbers. Let
\(k_1^2=\overline{a_n...a_1a_0}\) the
decimal representation of a square number.
Is it possible that \(\overline{a_n...a_1a_0a_0}\) and \(\overline{a_n...a_1a_0a_0a_0}\) are also
both square numbers?
I’m thinking of a positive number less than \(100\). This number has remainder \(1\) when divided by \(3\), it has remainder \(2\) when divided by \(4\), and finally, it leaves remainder \(3\) when divided by \(5\). What number am I thinking of?
I’m thinking of two prime numbers. The first prime number squared is thirty-six more than the second prime number. What’s the second prime number?
Does the equation \(9^n+9^n+9^n=3^{2025}\) have any integer solutions?
Let \(n\) be an integer (positive or negative). Find all values of \(n\), for which \(n\) is \(4^{\frac{n-1}{n+1}}\) an integer.
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 \(2025\)?
Prove that the product of five consecutive integers is divisible by \(120\).
For any positive integer \(k\), the factorial \(k!\) is defined as a product of all integers between 1 and \(k\) inclusive: \(k! = k \times (k-1) \times ... \times 1\). What’s the remainder when \(2025!+2024!+2023!+...+3!+2!+1!\) is divided by \(8\)?