3. [5] What is the largest $x$ such that $x^{2}$ divides $24 \cdot 35 \cdot 46 \cdot 57$ ?
Answer: 12 We factor the product as $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19 \cdot 23$. If $x^{2}$ divides this product, $x$ can have at most 2 factors of 2,1 factor of 3 , and no factors of any other prime. So $2^{2} \cdot 3=12$ is the largest value of $x$.