Question

A rectangular floor is 20 feet long and 16 feet broad. if it is to be paved with squared marbles of same size,find the greatest length of each squared marbles.​

Answer 1

4 ft

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.