Hey Steve,

Your next 2 questions are answered as follows:

Q1.

<Similar question appeared in NY 2013 Prelim>

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, 30, 36, 42

Odd Seated: 3, 9, 15, 21, 27, 33, 39

For multiple of 4 (every 4 pupils):

Even Stand: 6, 18, 30, 42, 4, 8, 16, 20, 28, 32, 40

Odd Seated: 3, 9, 15, 21, 27, 33, 39

For multiple of 5 (every 5 pupils):

Even Stand: 6, 18, 42, 4, 8, 16, 28, 32, 10

Odd Seated: 3, 9, 21, 27, 33, 39, 5, 25, 35

For multiple of 6 (every 6 pupils):

Even Stand: 4, 8, 16, 28, 32, 10, 12, 24, 30, 36

Odd Seated: 3, 9, 21, 27, 33, 39, 5, 25, 35

Based on the pattern,

1 is always standing;

Look at 3. It was first standing (after 2), seated (after 3) and remains seated. Its factors are 1, 3.

Same with 5, 6.

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.

Same with 9.

From 1 to 42, identify those numbers that have an odd number of factors.

These are ....

1, 4, 9, 16, 25, 36

Incidentally, all these are also square of numbers.

Hence **6** pupils were standing in the end.

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Q2.

<Qty Balloons>

S - B = 48

Given that each part of length 90cm(L90) had 5 small(S) balloons and each part of length 60cm(L60) had 2 big(B ) balloons, this infers that L90 part comes from the longer string (LS).

<length>

L90 : L60

90 : 60

3 : 2

Given Long String (LS) : Short String (SS) is 3 : 1, and that L90 part comes from the LS, multiply the ratio component of L90 by 2x, and this gives the 3 : 1 comparison. Thus we have ...

2 parts of L90 and 1 part of L60 that gives the 3 : 1 (LS : SS) comparison (in terms of its total length)

L90 : L60

2 : 1

In terms of length, it becomes ...

180 : 60

3 : 1

In terms of number of balloons, we have ...

L90(S) : L60(B )

10 : 2

This gives a difference of 8 balloons {8 fewer B than S}.

Given that this difference is 48, which is a multiple of 6, we have ...

L90(S) : L60(B )

60 : 12 {48 fewer B than S}

therefore, there were **72** balloons altogether {60 + 12}

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Trust these help.

Do let me know again if these are different from your <Anskey> or if there's further clarification.

Cheers,

Edward