GMAT Problem Solving- Find the greatest number that will divide 43, 91 and 183

Question: Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

1. 4
2. 7
3. 9
4. 13
5. 24

”Find the greatest number that will divide 43, 91 and 183”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

Based on the given question above that requires to find the greatest number that will divide 43, 91 and 183 leaving the same remainder in each case, five options for the right answer has been given. To find the correct answer, each case needs to be evaluated individually thereby dividing each of the options with each of the given case numbers. However, the greatest number is to be searched for hence, among the five options, it can be started with the last gradually moving towards the first one.

5. 24

When 43 is divided by 24, the result is 1 with remainder stands as 19
When 91 is divided by 24, the results is 3 and the remainder comes out as 19
When 183 is divided by 24, it equals 7 and the remainder received is 15.
Since the greatest number to be divided by all the three cases needs to endure the same remainder, 24 is not the right answer.

500. 13

When 43 is divided by 13, the result is 3 with remainder stands as 4
When 91 is divided by 13, the results is 7 and the remainder comes out as 0
When 183 is divided by 13, it equals and the remainder received is 11.
Since the greatest number to be divided by all the three cases needs to endure the same remainder, 13 is not the right answer.

100. 9

When 43 is divided by 9, it equals to 4 with remainder as 7
When 91 is divided by 9, it equals 10 with remainder as 1
When 183 is divided by 9, it equals to 20 with remainder 3
Since the remainder needs to be the same each time, the number 9 is not the correct answer.

2. 7

When 43 is divided by 7, it equals to 6 with remainder as 1
When 91 is divided by 7, it equals 13 with remainder as 0
When 183 is divided by 7, it equals to 26 with remainder 1
Since the remainder needs to be the same each time, the number 7 is not the correct answer.

1. 4

When 43 is divided by 4, it equals to 10 with remainder as 7
When 91 is divided by 4, it equals 22 with remainder as 7
When 183 is divided by 4, it equals to 45 with remainder 7
Since for all the three cases, the remainder achieved is 7, hence, option A with 7 is the right answer.
Therefore, the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case is 4.
Correct Answer: A

Approach Solution 2:

Another way to find out the greatest number to be divided with 43, 91 and 183, which will leave the same reminder, can be done with the Highest common Factorisation method.
For the purpose, it is important to calculate the HCF of each of the numbers by first subtracting all of these cases with another. This implies-

HCF
= 91-43 = 48
= 183-91 = 92
= 183-43 = 140

For each of these numbers including 48, 92 and 140, the Highest Common Factor stands as 4.
This implies that if 43, 91 and 183 is divided by 4, then the achieved remainder would be the same for each of these cases. Hence, 4 is the greatest number that will divide each of the cases.
Correct Answer: A

Approach Solution 3:

Let's look at the cases that need to be evaluated separately in order to answer this problem, then divide each alternative by each of the case numbers provided. The biggest number must be found, though, so among the five choices, it may be begun with the last and worked its way up to the first.

1.  If you divide 43 by 4, you get 10 with a 7 as the remainder.
   91 divided by 4 gives 22 and 7 as the remaining.
   When 183 is divided by 4, the result is 45, leaving a 7-point residual.
   Since the obtained remainder in each of the three scenarios is 7, option A with 7 is the correct response.
   In order to leave the same residual in each case when dividing 43, 91, and 183, the largest integer that will do so is 4.

Correct Answer: A

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