1. [15] A positive integer $k$ is called powerful if there are distinct positive integers $p, q, r, s, t$ such that $p^{2}, q^{3}, r^{5}, s^{7}, t^{11}$ all divide $k$. Find the smallest powerful integer.
Answer: 1024
First of all, 1024 is powerful because it can be divided evenly by $16^{2}=256,8^{3}=512,4^{5}=1024,2^{7}=$ 128 , and $1^{11}=1$.
Now we show that 1024 is the smallest powerful number. Since $s \neq t$, at least one of them is at least 2 . If $t \geq 2$ or $s \geq 3$, then we need the number to be divisible by at least $2^{11}=2048$ or $3^{7}=2187$, which both exceed 1024, so we must have $s=2$ and $t=1$. If $r=3$, then the number must be divisible by $3^{5}=243$ and $2^{7}=128$, which means that the number is at least $243 \cdot 128>1024$, so $r \geq 4$, and the number is at least $4^{5}=1024$. Therefore the smallest powerful number is indeed 1024 .