4. Imagine you have a floor that measures 11,025 by 630,630...

4. Imagine you have a floor that measures 11,025 by 630,630 (the units do not matter), and you wish to tile it using squares. However, you are not allowed to break any tiles, and all the tiles you use must be the same size. (a) Explain why 1 !! 1 tiles will work. (b) Explain why 2 !! 2 tiles will not work. (c) Explain why 3 !! 3 tiles will work. (d) What is the largest square tile that will work? Do not quote a formula!

Transcript

00:03 Hi.

00:04 So one inch by one inch tiles would work here.

00:08 And the reason for that is because both 11 ,025 and 630 ,630 are divisible by one.

00:19 So in other words, i would have 11 ,025 tiles going this way and 11 ,025 tiles or i'm sorry, and 630 ,630 tiles going down this way.

00:31 So now let's think about, i'm going to take this away.

00:34 If we have two by two tiles, so two units by two units.

00:40 Well, two is not, or sorry, 11 ,025 is not divisible by two.

00:52 So that means i'm going to have some leftover tile here, and i can't cut it.

00:58 So if i have two by two tiles, it will not work because 11 ,025 is not divisible by two.

01:04 It would work this long way because it is 630 ,630 is divisible by 2, but we would have to cut the edges on this side, and we're not supposed to do that.

01:17 All right.

01:18 So now we're looking at three by three tiles.

01:23 A good way to check divisibility for 3 is to add up all the digits in the number and see if that's divisible by 3.

01:30 So 1 plus 1 plus 0 plus 2 plus 5.

01:33 So 1 plus 1 is 2.

01:35 Plus two is four, plus five is nine, and since three fits into nine, three will fit into 11 ,025.

01:43 Let's check 630 ,630 .6 plus three is nine, and six plus three is nine, and nine plus nine is 18.

01:51 18 is divisible by three.

01:54 So this number would also be divisible by three.

01:56 So yes, three by three tiles would work because both numbers are divisible by three.

02:07 So i'm going to get rid of this just so that i have space now.

02:11 The last part of your question is asking, what is the biggest square tile that would work without having to cut them? so basically, this is really a factoring question...

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