Suppose that \(n\) is a natural number and \(p\) is a prime number. How many numbers are there less than \(p^n\) that are relatively prime to \(p^n\)?
Find the minimal natural number \(n>1\) such that \(n^6 - 2n^5 - n^4 + 4n^3 - n^2 - 2n +1\) is divisible by \(2025\).
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 \(1 + 2 + ... + n = nm\).
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?