The greatest of all

The idea

The Highest Common Factor (HCF) of two or more numbers is the highest (greatest) of their common factors; it is also called the Greatest Common Divisor (GCD).

Tiles for a floor

Sameeksha’s room is 12 ft by 16 ft. She wants square tiles of one size with no cutting: the tiles must fit the 12 ft and the 16 ft exactly.

Which tile side do you think fits both the 12 ft and the 16 ft exactly?

• 4 ft
• 5 ft
• 8 ft

Slide the tile size

Slide the size. A red strip means the tiles do not fit. Collect every size that fits, then lock the size that needs the fewest tiles.

The biggest common factor

For tiles to fit the breadth 12 ft, the side must be a factor of 12. For the length 16 ft, it must be a factor of 16. So the side is a common factor of 12 and 16.

The floor has the same area whatever the tile. Bigger tiles cover more each, so fewer are needed: 1 ft needs 192 tiles, 2 ft needs 48, and 4 ft needs only 12. So she takes the largest common factor, 4 ft.

The Highest Common Factor (HCF) of two or more numbers is the highest (greatest) of their common factors. It is also called the Greatest Common Divisor (GCD). Here the HCF of 12 and 16 is 4.

• Number | Factors
• 12 | 1, 2, 3, 4, 6, 12
• 16 | 1, 2, 4, 8, 16
• Common | 1, 2, 4

Notes

The Highest Common Factor (HCF) of two or more numbers is the highest (greatest) of their common factors, also called the Greatest Common Divisor (GCD).

Check yourself

Two ropes, 18 m and 30 m long, are cut into pieces of equal length, with no rope left over. The pieces are as long as possible. How many metres long is each piece?

Answer: 6 m

Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Common: 1, 2, 3, 6. The highest is 6 m.

What is the HCF of 8 and 12?

Which is the HCF of 15 and 40?

A hall is 24 ft long and 36 ft broad. It is covered with square tiles of the biggest side that fits exactly. How many tiles are used?

Answer: 6

HCF of 24 and 36 is 12. Along 24 ft there are 2 tiles and along 36 ft there are 3 tiles: 2 × 3 = 6 tiles.

• 2. 2 is a common factor of 8 and 12, but it is not the highest: 4 divides both too.
• 4 — correct. Yes! The common factors are 1, 2 and 4, and 4 is the highest.
• 24. 24 is a common multiple of 8 and 12. A factor of 8 cannot be bigger than 8.
• 5 — correct. Yes! The factors of 15 are 1, 3, 5, 15. Of these only 1 and 5 divide 40. The highest is 5.
• 10. 10 divides 40, but 10 does not divide 15, so it is not a common factor.
• 15. 15 divides 15, but 15 does not divide 40, so it is not a common factor.