Is 9 the HCF of p and q? GMAT Data Sufficiency

Question: Is 9 the HCF of p and q?

1) LCM of p and q = 75
2) pq = 225

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
1) Since 75 is not a multiple of 9, 9 cannot be HCF
Statement 1 is sufficient.

2) pq = 225
If 9 is HCF the number can be 9a or 9b the product pq = 81ab
Since 225 is not a multiple of 81.
Statement 2 is also sufficient.

Approach Solution 2:
The question becomes simple if below important property is noted:
The HCF of a group of numbers is always a factor of their LCM.

1. HCF is the product of all common prime factors using the least power of each common prime factor
2. LCM is the product of the highest powers of all prime factors.

Statement 1: LCM of two numbers is 75. Since, 9 is not a factor of 75 (3x5x5), 9 cannot be HCF of the two numbers. (Possible HCF could be: 3, 5, 15, 25, 75) Sufficient
Statement 2: HCF should be a factor of Product of two numbers. 9(3x3) is a factor of 225 (3x3x5x5). But so are 15 (e.g. for p=15 , q=15), 25, 45, 75.
Also, for HCF to be 9, the product of numbers should be divisible by 9 twice i.e. 81. Once as a factor of p and then as a factor of q. Since, 225 is not divisible by 81. Sufficient

Approach Solution 3:
(1) LCM (p ; q) = 75 = (3) (5)^2

If either P or Q had 9 as a factor, then it would NOT evenly divide into the LCM ——> and each number in the set, by definition of the LCM being a multiple of every number in the set, must divide evenly into the LCM
So we have a definite NO because any number that is divisible by 9 can never divide into 75. Therefore, P and Q can NOT share a common factor of 9.

Another way to look at it is that the LCM of a set of numbers always consist of: (GCF) * (remaining coprime factors that remain in each number)

Assuming the GCF of P and Q were = 9
Then:

P = 9(a) ——— and ———- Q = 9(b)
Where a and b are COPRIME Integers - because P and Q have no more common factors

The LCM in this case would be:
LCM = (GCF) * (a * b)
LCM = (9) * (a * b)

Since 225 is not divisible by 9, the GCF of p and q can NOT be 9

S1 sufficient
S2: p * q = 225

When we multiply P and Q, we are essentially “combining” all the prime factors of each number into one larger number.
Therefore, in order for P and Q to share a common factor of 9, at the very least the product of the 2 integers must be divisible by:
P * Q = (9a) (9b) = 81 (a) (b)

Since 225 is not divisible by 81, it is impossible for P and Q to share a GCF of 9
S2 sufficient

“Is 9 the HCF of p and q?”- is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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