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Secondary 1 | Maths

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(Prime factorisation)

1000 = 2³ × 5³

You can write 1000 as the product of two numbers in a few ways:

1 x 1000

2¹ x (2 x 5³) = 2 x 500

2² x (2 x 5³) = 4 x 250

5 x (2³ x 5²) = 5 x 200

2³ x 5³ = 8 x 125

(2 x 5) x (2² x 5²) = 10 x 100

(2² x 5) x (2 x 5²) = 20 x 50

5² x (2³ x 5) = 25 x 40

1st student opens door 1000

2nd student closes it

4th student opens it

5th student closes it

...

...

500th student opens it.

1000th student closes it.

When written as the product of two whole numbers, there is an even number of unique factors.

So for every time the door is open/closed, it will be reversed.

If a number has an even number of unique factors, it will be in the closed state in the end.

Example: door 64

64

= 1 x 64

= 2 x 32

= 4 x 16

= 8 x 8 = 8²

We notice that 8 is repeated so it doesn't have another unique factor to pair with it.

So the door's state is changed an odd number of times. It will remain open in the end.

The doors that remain open are:

1² = 1

2² = 4

3² = 9

4² = 16

...

...

...

31² = 961