# How many multiples of 6 are possible

### Flight operations

At an airport, a plane leaves for London every 3 hours. A plane arrives from Warsaw every 5 hours. It happens that aircraft on these two routes are parked in adjacent places. An airplane just took off and one landed.

After how many hours are two aircraft on these lines next to each other again?

(You only need to examine the next 60 hours. :-))

- After how many hours do the planes leave for London?

These are the multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60. - After how many hours does a plane land in truth?

These are multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60.

The planes meet at the numbers that appear in both multiple rows.

So: 15, 30, 45 and 60.

After 15 hours, the planes on the two routes are side by side again for the first time.

The 15 hours are that **least common multiple** (kgV). Most of the time, that's the only thing that interests you **smallest** common multiples, not some common multiple.

If you write down the multiples of 2 numbers, this is the first common number **least common multiple** (kgV).

Example: Lcm (3; 5) = 15

Image: Fraport AG photo team (Stefan Rebscher)

### Example for the LCM

Find the least common multiple of 6 and 8.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...

Multiples of 8: 8, 16, 24, 32, 40, ...

There was already the LCM: 24

Lcm (6; 8) = 24

### Furniture manufacturing

A factory wants to make cabinets that are 120 cm or 150 cm high. Both cabinets should have drawers. And the drawers should have the same height, which is easier for the production process.

How high can the drawers be?

There always have to fit all the drawers in the closet. So you think about the numbers by which the cabinet height can be divided. These are the divisors of 120 and 150.

- Divisors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
- Divisors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

The numbers that appear in both rows fit as drawer height in both cabinets. Well, 2 cm in height would be nonsense. Look for the greatest divisor. That is 30. 30 cm fits well as a drawer height.

30 is the **greatest common factor** from 120 and 150.

If you write down the divisors of 2 numbers, the greatest common number is the **greatest common factor** (gcd).

Example: gcd (8; 12) = 4

Image: fotolia.com (fotandy)

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### Example for the GCD

Find the greatest common factor (common factor) of 28 and 42.

Divisors of 28: 1, 2, 4, 7, 14, 28.

Divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The gcd is: 14

gcd (28; 42) = 14

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