Answer:
4 ft
Step-by-step explanation:
I guess, the meaning is the largest marbles, so that we can pave the whole floor without cutting any marbles and leaving empty spots.
so, 20×16 ft²
we can have marbles 1/2 ft long. and it all fits well : 40×32 marbles.
we can have them 1 ft long, and it all fits well : 20×16 marbles.
we can have them 2ft long, and it still fits well : 10×8 marbles.
and so on.
so, actually, we are looking for the greatest common divisor (GCD) of 20 and 16. and that gives us the maximum length of a single marble to fulfill the requirement.
let's go for the prime factors starting with 2
20/2 = 10
10/2 = 5
5/3 fits not work
5/5 = 1 done
so, 20 = 2²×3⁰×5¹
16/2 = 8
8/2 = 4
4/2 = 2
2/2 = 1 done
16 = 2⁴
so, for the GCD I can only use powers of 2 (the only prime factors both numbers have in common).
and we have to use the smaller power of 2, which is 2, so, the GCD is 2² = 4
=>
the maximum length of the squared marbles is 4 ft.
that would pave the floor with 5×4 marbles completely.
