A Colony has Houses Numbered 1 to 150 GMAT Problem Solving

Question: A colony has houses numbered 1 to 150. Three guards, Tim, Jack & John, were appointed by the colony in charge. Tim was to protect each house whose number is a multiple of 2 and 5, Jack was to protect each house whose number is a multiple of 2 and 3 whereas John was asked to protect rest of the houses. How many houses were to be protected by both Tim & Jack?

1. 2
2. 3
3. 4
4. 5
5. More than 5

Correct Answer: D

Solution and Explanation:

Approach Solution 1:

Given that houses in a colony range from 1 to 150. The colony in charge appointed Tim, Jack, and John as three guards. Tim was tasked with guarding each house whose number was a multiple of 2 and 5, Jack with guarding each house whose number was a multiple of 2 and 3, and John with guarding the other houses.

Tim guards houses that are multiples of 2 and 5, meaning house no.s 10, 20, so on.
Multiples of 10.
Similarly, Jack guards houses that are multiples of 2 and 3, meaning house no.s 6, 12, 18, so on.
Multiples of 6.
Question is- How many houses did they guard together.
That would be given by the houses that are both the multiples of 10 and 6.
That can be given by LCM (10,6) = 30.

So there will be 150 / 30 = 5 houses that will be guarded by both Tim and Jack together.

Approach Solution 2:

Given that houses in a colony range from 1 to 150. The colony in charge appointed Tim, Jack, and John as three guards. Tim was tasked with guarding each house whose number was a multiple of 2 and 5, Jack with guarding each house whose number was a multiple of 2 and 3, and John with guarding the other houses.

Now we know that Tim guards the houses that are multiple of 2 and 5.

The house numbers will be multiple of both 5 and 2

LCM(5,2) = 10
The houses numbers will be 10, 20, 30, 40, 50
Jacl guards the houses that are multiple of 2 and 3.
The house numbers will be multiple of both 3 and 2
LCM(3,2) = 6
The houses numbers will be 6, 12, 18, ..
The houses which will be common in them has to be counted twice. Those houses will be multiple of 6 as well as 10
LCM(6,10) = 30
Total common houses = 150 / 30 = 5

“A colony has houses numbered 1 to 150. Three guards, Tim, Jack & John, were appointed by the colony in charge" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”. To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. The problem-solving section of the GMAT Quant topic is made up of very complicated maths problems that must be solved by using the right maths facts.

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