- This is Problem 8 from the 2015 Fermat Contest.
If \(3^x5^y=225\), determine the value of \(x+y\).
- Find the GCF and LCM of two numbers written as \(m^3n^2pq\) and \(mn^3qr\), where \(m,n,p,q,r\) are prime.
- This is Problem 20 from the 2017 Fermat Contest.
If \(m\) and \(n\) are positive integers with \(n\gt1\) such that \(m^n=2^{25}\times3^{40}\), determine the value of \(m+n\).
- Find the value of positive integers \(n\) and \(y\) such that \(4^{20}\times5^{30}=2^n\times10^y\).
- \(7n\) is the square of a number and \(784np\) is the cube of a number, where \(n\) and \(p\) are positive integers. What is the smallest value of \(n+p\)?
- Determine the smallest positive integer \(k\) such that \(2160k\) has both a square root and a cube root that are both integers.
- For every positive integer \(n\), \(n!\) (read "n factorial") represents the product of the first \(n\) positive integers.
For example,
\(\begin{align*} 8!&=8\times7\times6\times5\times4\times3\times2\times1\\ &=(2\times2\times2)\times7\times(2\times3)\times5\times(2\times2)\times3\times2\times1\\ &=2^7\times3^2\times5\times7 \end{align*}\)
Determine the value of \(n\) for which \(n!=2^{25}\times3^{13}\times5^6\times7^4\times11^2\times13^2\times17\times19\times23\).