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A certain number of cars were arranged by a hotel to escort its customers to the party venue. If 4 people sit in a car, the rest of them would have to divide themselves in groups of either 6 people in each car or 9 people in each car. Since only small cars were available, hotel management arranged the travel by sending 5 people in each car and the remaining had to take a taxi. What is the minimum number of people who took taxi if the total number of people were greater than 40 and less than 60?

A. 0
B. 1
C. 2
D. 3
E. 4

My Approach:

So at first you feel question says the number of people can be >40 and <60 and it has to be divided in group of 5. So It can be 45,50, or any other number leaving the question ambiguous.
So let's try to find the number of people from information prior to it: If I subtract 4 from number the number is multiple of both 6 and 9. So if you think of it 6*9+4 itself results in 58 and is between 40 and 60.
Which means your number is 58.
So if 58 is divided by 5 the remainder is 3.
Hence D.
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If 4 people are placed in one car, the remaining will have to be transported either in groups of 6 or 9, in other cars. Translation: The remaining number of people, \( X\), is a multiple of both 6 and 9. Since the LCM of 6 and 9 is 18, \(X\) will be a multiple of 18.

\(TotalPeople = 4 + X\). We are also told that total number of people was greater than 40 and less than 60.

Thus,

\(40 < 4 + X < 60 \\
36 < X < 56\)

We now have two pieces of information about X: One that it is between 36 and 56 and second, that it is a multiple of 18.

18 * 1; Not possible because X is between 36 & 56
18 * 2 = 36; Not possible because X is between 36 & 56, exclusive.
18 * 3 = 54; Possible.

Thus total number of people = 54 + 4 = 58.

When divided by 5, 58 leaves a remainder of 3. Thus D is the answer.
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A certain number of cars were arranged by a hotel to escort its customers to the party venue. If 4 people sit in a car, the rest of them would have to divide themselves in groups of either 6 people in each car or 9 people in each car. Since only small cars were available, hotel management arranged the travel by sending 5 people in each car and the remaining had to take a taxi.

What is the minimum number of people who took taxi if the total number of people were greater than 40 and less than 60?

The number of customers = 4 + LCM(6,9)*k = 4 + 18k = 5m + r

40 < 4 + 18k < 60
36 < 18k < 56
2 < k < 3.11; k = 3

The number of customers = 4 + 18k = 4+18*3 = 58 = 5m + r; m = 11; r = 3

IMO D
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