Club X has more than 10 but fewer than 40 members. Sometimes the members

BTGModeratorVI wrote: ↑

Wed Feb 03, 2021 10:37 am

Club X has more than 10 but fewer than 40 members. Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables, and sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables. If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer: E
Source: official guide

This is a nice remainder question in disguise.
For this question, we'll use a nice rule that that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Let N = the TOTAL number of members.

Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables...
With 4 members at each table, then N is multiple of 4
However, we still have one more table to consider.
Since the last table has 3 members, we know that N is 3 greater than a multiple of 4
In other words, when we divide N by 4, the remainder is 3
By the above rule, some possible values of N are: 11, 15, 19, 23, 27, etc
NOTE: I started at 11, since we're told that 10 < N < 49

Sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables
Using the same logic as above, this question tells us that, when we divide N by 5, the remainder is 3
By the above rule, some possible values of N are: 13, 18, 23, 28, 33, 38

Let's check the two results.
First we learned that N can equal 11, 15, 19, 23, 27, 31, 35, 38
Next we learned that N can equal 13, 18, 23, 28, 33, 38
Once we check the OVERLAP, we can see that N equals 23

If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members?
If N = 23, then we'll have 3 tables with 6 members and the remaining 5 members will sit at the other table.

Answer: E

BTGModeratorVI wrote: ↑

Wed Feb 03, 2021 10:37 am

Club X has more than 10 but fewer than 40 members. Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables, and sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables. If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer: E
Source: official guide

Solution:

Although this problem appears to be a general word problem, it is actually testing us on our understanding of remainders when dividing integers. We are first told that the total number of members, which we can denote as T, is between 10 and 40. Next, we are told two important pieces of information:

1) “Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables.”

2) “Sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables.”

Let’s now translate these into two mathematical expressions.

1) T/4 = Quotient + Remainder 3

2) T/5 = Quotient + Remainder 3

Because the remainder (3) is the same when T is divided by 4 or by 5, the remainder will still be 3 when T is divided by the LCM of 4 and 5. That is, we are really looking for the following:

T/20 = Quotient + Remainder 3.

Since T must be between 10 and 40, there is only one value in that range that produces a remainder of 3 when divided by 20. That value is 23. We can now use this value to complete the question. We are finally asked:

“If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members?”

This is same as asking the following: what is the remainder when 23 is divided by 6? We can see that 6 divides into 23 three times with a remainder of 5.

Answer: E