Sameeksha's floor tiles

Sameeksha is building her new house. The main room is 12 ft by 16 ft, and she wants the floor covered with square tiles of the same size. She wants to use as FEW tiles as possible — and fewer tiles means each tile must be bigger — and the side of the tile to be a whole number of feet. What size should she buy?

Think about what must be true of the tile's side. For the tiles to fit the breadth exactly, with none cut, the side must divide 12 — it must be a FACTOR of 12. For them to fit the length exactly, it must be a factor of 16.

So the side must be a factor of BOTH. List them. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 16 are 1, 2, 4, 8, 16.

Comparing the two lists, the COMMON factors are 1, 2 and 4. So square tiles of side 1 ft, 2 ft or 4 ft would all fit the room exactly.

Which should she choose? She wants as few tiles as possible, and a bigger tile covers more floor, so fewer are needed. The largest of the three is 4, so she needs tiles of side 4 ft.

The Highest Common Factor (HCF) of two or more numbers is the highest of their common factors. It is also known as the Greatest Common Divisor (GCD); in Hindi it is महत्तम समापवर्तक. Here, 4 is the HCF of 12 and 16.

Two questions worth asking. How many 4 ft tiles does she need? And what if the side did not have to be a whole number of feet — would the answer change? (It would: with fractional sides there is no largest tile, so the 'whole number' condition is doing real work.)