For any positive integer \(k\), the factorial \(k!\) is defined as a product of all integersbetween 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 \(8\)?
Find all functions \(f\) from the real numbers to the real numbers such that \(xy=f(x)f(y)-f(x+y)\) for all real numbers x and y.
There are two imposters and seven crewmates on the rocket ‘Plus’. How many ways are there for the nine people to split into three groups of three, such that each group has at least two crewmates? The two imposters and seven crewmates are all distinguishable from each other, but we’re not concerned with the order of the three groups.
For example: \(\{I1,C1,C2\}\), \(\{I2,C3,C4\}\) and \(\{C5,C6,C7\}\) is the same as
\(\{C3,C4,I2\}\), \(\{C5,C6,C7\}\) and \(\{I1,C2,C1\}\) but different from
\(\{I2,C1,C2\}\), \(\{I1,C3,C4\}\) and \(\{C5,C6,C7\}\).
Let \(n\) be a natural number, and let \(d(n)\) be the number of factors of \(n\). For example, the factors of \(6\) are \(1,2,3,6\), so \(d(6)=4\). Find all \(n\) such that \(d(n)+d(n+1)=5\).