How many factors does positive integer z have?(1) z/5 and z/7 and are integers and the greatest integer that divides them both is 8(2) The smallest integer that is divisible by both z and 14 is 280a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer?

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How many factors does positive integer z have?
(1) z/5 and z/7 and are integers and the greatest integer that divides them both is 8
(2) The smallest integer that is divisible by both z and 14 is 280

a)
Statement (1) ALONE is sufficient, but statement (2) alone is
not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is
not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to
answer the question asked, but NEITHER statement ALONE
is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question
asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to
answer the question asked, and additional data specific to the
problem are needed.

Correct answer is option 'A'. Can you explain this answer?

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How many factors does positive integer z have?(1) z/5 and z/7 and are ...

Step 1 & 2: Understand Question and Draw Inference

Given: Integer z > 0
To find: Number of factors of z

Let z = P *P *P * . . . , where P , P , P etc. are prime numbers and a, b, c . . . are non-negative integers
So, number of factors of z = (a+1)*(b+1)*(c+1)* . . .
So, in order to be able to apply the formula for number of factors, we need to know the prime-factorized form of z

Step 3 : Analyze Statement 1 independent

Statement 1 says that ‘z/5 and z/7 and are integers and the greatest integer that divides them both is 8’

Since both z/5 and z/7 and are integers, this means, z contains 5 and 7 for sure

Upon comparing the form 5^a-1*7^b-1 * P₃^c* . . . with 2³ (which can be rewritten as 2³ *5⁰ *7⁰ for easier comparison) , we see that:
- P₃ = 2 and c = 3
- a – 1 = 0; So, a = 1
- b – 1 = 0; So, b = 1
- And z has no other prime factor (else it would have appeared in the GCD of z/5 and z/7
- So, z = 2³*5*7
- Since we now know the prime-factorized form of z, we can answer the question
- Statement 1 is sufficient

Step 4 : Analyze Statement 2 independent

Statement 2 says that ‘The smallest integer that is divisible by both z and 14 is 280’

So, LCM(z, 14) = 280
- 280 = 2³*5*7
14 = 2*7
The power of 2 in 14 is only 1. Therefore, the 2 term in LCM(z, 14) must come from z.
14 does not contain any 5. So, the 5 in LCM(z, 14) must come from z
14 contains 7¹, same as the LCM(z, 14) does. So, the power of 7 in z may be 0 or 1
Therefore, either z = 2³*5*7⁰ or z = 2³*5*7¹
Correspondingly, the number of factors of z = Either (3+1)*(1+1)*(0+1) or (3+1)*(1+1)*(1+1)
Since Statement 2 leads us to two possible values for the number of factors of z, it is not sufficient to get a unique answer

Step 5: Analyze Both Statements Together (if needed)

Since we’ve already arrived at a unique answer in Step 3, this step is not
required
Answer: Option A

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How many factors does positive integer z have?(1) z/5 and z/7 and are integers and the greatest integer that divides them both is 8(2) The smallest integer that is divisible by both z and 14 is 280a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer?

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How many factors does positive integer z have?(1) z/5 and z/7 and are integers and the greatest integer that divides them both is 8(2) The smallest integer that is divisible by both z and 14 is 280a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about How many factors does positive integer z have?(1) z/5 and z/7 and are integers and the greatest integer that divides them both is 8(2) The smallest integer that is divisible by both z and 14 is 280a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many factors does positive integer z have?(1) z/5 and z/7 and are integers and the greatest integer that divides them both is 8(2) The smallest integer that is divisible by both z and 14 is 280a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer?.

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