Question

hannah is making a game board that has a dimension of 15 inches by 24 inches if she will use square tiles what is the largest tile she can use

Answer 1

If square tiles with different sizes are allowed it follows that the largest tile she can use is $15\times 15.$

If different square tiles must have the same sizes, it follows that the largest tile she can use is $gcd(15,24)\times gcd(15,24)=3\times 3,$ where $gcd(15,24)$ is the greatest common divisor of 15 and 24.