The “Connection” Between Any Two Positive Integers a And b GMAT Problem Solving

Question: The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b. For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1. How many possible values of y are there?

1. 7
2. 8
3. 9
4. 10
5. 11

“The “connection” between any two positive integers a and b''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The GMAT quant section determines the calculative skills of the candidates. The candidates must estimate the sum logically to find the correct option. The candidates must have a solid knowledge of arithmetic, algebra and interpreting graphical data to crack GMAT Problem Solving questions. The mathematical problems of the GMAT Quant topic in the problem-solving part can only be solved with better calculative knowledge. The candidates can polish up their mathematical skills questions by answering from the book “GMAT Official Guide Quantitative Review 2019”.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b
• The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1.

Find Out:

• The number of values of y.

Since the problem statement states that the “connection” between y and 6 is 1/1, then LCM of (6, y) is equal to 6y. This is because the numbers 6 and y are co-prime numbers. These numbers do not have any common factor other than 1. Therefore, the LCM (6, y) would be less than 6y.

It is required to check the number of integers less than 20 is co-prime with 6. This means it is required to check the number of integers less than 20 is not divisible by 2 or 3 (since 6 is the product of 2 and 3).
Therefore, there exists (18-2)/2+1=9 multiples of 2 within the range between 0 and 20 that is not inclusive.

There exists (18-3)/3+1=6 multiples of 3 within the range between 0 and 20 that is not inclusive.
There exist 3 multiples of 6 within the range between 0 and 20, not inclusive (6, 12, 18)- the overlapping of the above two sets.

Therefore, absolute multiples of 2 or 6 within the range from 0 to 20, not inclusive is 9+6-3=12
Absolute integers within the range from 0 and 20 are 19

Therefore, there is an aggregate of 19-12=7 numbers which does not have any common factor with 6 other than 1: 1, 5, 7, 11, 13, 17 and 19.

Correct Answer: (A)

Approach Solution 2:

The problem statement suggests that:

Given:

• The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b
• The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1.

Find Out:

• The possible number of values of y.

The question suggests that the “connection” between two positive integers a and b is the ratio of LCM of a and b to the product of a and b.
The ratio of y and 6 is 1:1, therefore we can say that the LCM and products of the numbers are equal.
This can occur only if the numbers y and 6 possess no common factor other than 1.
Since the number 6 consists of two factors 2 and 3, the values of y can be all those numbers that are not the multiples of 2 and 3 and less than 20.
Therefore, y= (1,5,7,11,13,17,19)

Thus we can infer that there is a total of 7 possible values of y.

Correct Answer: (A)

Approach Solution 3:

The problem statement indicates that:

Given:

• The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b
• The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1.

Asked:

• To find the possible number of values of y.

According to the formula, the product of two numbers is equal to the product of HCF and LCM.
Since the connection between y and 6 is equal to 1/1,
Then we can say, LCM/ (LCM* HCF) is equal to 1
Or, HCF= 1
Therefore, we are searching for the co-primes to 6.
The factors of 6 are 2 and 3.
Thus the values of y can not be the multiples of 2 and 3 and will be less than 20.
Therefore, the possible values of y are 1,5,7,11,13,17,19

Hence, there are 7 possible values of y.

Correct Answer: (A)

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