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 By: geniuskids (offline)  Saturday, September 14 2013 @ 10:58 PM CDT (Read 1534 times)
geniuskids

Hi there, would appreciate your help on attached. Q18 (b) of Nanyang Paper

Q18 (a) 51 digits (managed to get answer)
(b) 5 pupils (need help on this)

Thanks

Active Member

Registered: 11/12/11
Posts: 169

 By: echeewh (offline)  Monday, September 16 2013 @ 09:16 PM CDT
echeewh

Hey <

Following is my worked solution :

(a)
List it out:

1 - 9 = 9 numbers = 9 digits
10 - 30 = 21 numbers
21 × 2 = 42 digits

Total digits = 9 + 42 = 51

(b)
Apply the knowledge of Factors and/or Multiples to solve this.

Start by listing it out and analyze the pattern.

For multiple of 2 (every 2 pupils):
Seated: ALL Even
Stand: ALL Odd

For multiple of 3 (every 3 pupils):
Even Stand: 6, 12, 18, 24
Odd Seated: 3, 9, 15, 21, 27

For multiple of 4 (every 4 pupils):
Even Stand: 6, 18, 4, 8, 16, 20, 28
Odd Seated: 3, 9, 15, 21, 27

For multiple of 5 (every 5 pupils):
Even Stand: 6, 18, 4, 8, 16, 28, 10, 30
Odd Seated: 3, 9, 21, 27, 5, 25

For multiple of 6 (every 6 pupils):
Even Stand: 4, 8, 16, 28, 10, 12, 24
Odd Seated: 3, 9, 21, 27, 5, 25

Based on the pattern,

1 is always standing;

Look at 12. It was first seated (after 2), stand (after 3), seated again (after 4).
12 is multiple of itself (12), 2, 3, 4, 6. And its factors are 1, 2, 3, 4, 6, 12.

This infers that a number that has an even number of factors will be seated.

Conversely speaking, a number that has an odd number of factors will be standing.

Take a look at 4. Factors of 4 are 1, 2, 4. So pupil 4 will be standing in the end.

From 1 to 30, identify those numbers that have an odd number of factors.

These are ....

1, 4, 9, 16, 25.

Hence 5 pupils were standing in the end.

=======

Trust this helps.

Do let me know again if there's further clarification.

Cheers,
Edward

Active Member

Registered: 04/21/11
Posts: 627

 By: nisalam (offline)  Tuesday, September 17 2013 @ 04:42 AM CDT
nisalam

I think it is also safe to say that only numbers that are squares have an odd number of factors, so all you have to do in the end is look out for the squares in the list!

Junior

Registered: 04/29/10
Posts: 30

 By: geniuskids (offline)  Tuesday, September 17 2013 @ 07:01 AM CDT
geniuskids

Looks like I still have difficulty understanding. Don't seem to know what they are talking about!!!

Active Member

Registered: 11/12/11
Posts: 169

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