Bunuel wrote:

To rent an office, each member of a club must pay n dollars. If two more members join the club, the per-member payment would be reduced by two dollars. Which of the following could be the number of members currently in the club?

I. 16

II. 17

III. 18

A. I only

B. II only

C. I and III only

D. II and III only

E. I, II, and III

I went off sheer intuition here and tested numbers for Option I, then saw a pattern. The answer I got: all three are possible. Please correct me if I am mistaken.

I. 16

If there are 16 members, add 2 (=18), and multiply 16 * 18.

Do not find LCM; we need the product of 16 and 18 so that when we divide the product (total $) by (16 + 2), we will get $16.

16 members * $18 ea = $288

18 members would pay $288/18 = $16 each

16 members pay $18 each

18 members pay $16 each

2 more members, $2 less per person

II. 17

If there are 17 members, add 2 and multiply 17 * $19 = $323

19 members will pay $323/19 = $17 each

17 members pay $19 each

19 members pay $17 each

2 more members, $2 less per person

III. 18

Add 2. Multiply 18 * $20 ea = $360

20 members would pay $360/20 = $18 each

18 members pay $20 each

20 members pay $18 each

2 more members, $2 less per person

All three are possible.

Answer E

*

I started algebraically, but its implications befuddled me, so I switched to testing numbers.

A * n = S, where S must remain the same

A = amount paid per person

n = number of members

x = $ per person (i.e., is A)

Original:

x * n = xn = SUM

New

(x - 2)(n + 2) = same SUM

xn = (x - 2)( n + 2)

xn = xn + 2x - 2n - 4

4 = 2x - 2n

4 = 2(x - n)

2 = x - n

n + 2 = x

Don't laugh too hard if I'm over the cliff. I think this means the same as what I did instinctively: add 2 to the original number of members to get a dollar amount for what each pays originally.

That dollar amount turns out to be the new number of total members.
_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"