GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 17 Jul 2018, 16:28

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# A positive integer n has the smallest 3 prime number

Author Message
TAGS:

### Hide Tags

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1755
A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

Updated on: 28 Feb 2017, 04:21
2
8
00:00

Difficulty:

85% (hard)

Question Stats:

46% (01:48) correct 54% (01:32) wrong based on 122 sessions

### HideShow timer Statistics

Q. A positive integer n has the smallest 3 prime numbers as its only prime factors. How many positive integers divide n completely?

(1) The total number of times the prime factors of n occur in n is 5.
(2) The product of the number of times each prime factor of n occurs in n is 4.

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.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Thanks,
Saquib
Quant Expert
e-GMAT

Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

_________________

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Originally posted by EgmatQuantExpert on 28 Feb 2017, 04:20.
Last edited by EgmatQuantExpert on 28 Feb 2017, 04:21, edited 1 time in total.
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1755
A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

Updated on: 27 Mar 2017, 05:42
The official solution has been posted. Looking forward to a healthy discussion..
_________________

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Originally posted by EgmatQuantExpert on 28 Feb 2017, 04:20.
Last edited by EgmatQuantExpert on 27 Mar 2017, 05:42, edited 1 time in total.
Intern
Joined: 02 Feb 2017
Posts: 4
Re: A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

07 Mar 2017, 07:24
2
Given,

smallest three prime numbers are its ONLY prime factors, i.e, 2,3 & 5 . (Although we don't need to know what the factors are, just how many are there)

To find the number of integers that are factors of an integer n we need to know the power of prime factors.

Lets say n = 2^a * 3^b * 5^c

number of integers that are factors of n will be (a+1)(b+1)(c+1)

From 1st statement,
a+b+c = 5

Since we know none of a,b,c can be zero, lets assume a = 1. This gives following values for a,b and c:
1. a=1, b=2, c=2
2. a=1, b=3, c=1

So the # of factors can be = 16 or 18

Not sufficient.

From statement 2, a*b*c = 4. Again fixing a=1 we get following values of a,b,c
1. a=1,b=2,c=2
2. a=1,b=4,c=1

Hence, # of factors = 18 or 20

Hence, insufficient.

From 1 and 2, we can get that the number of factors are 18. Hence, sufficient.

Senior Manager
Joined: 24 Apr 2016
Posts: 333
Re: A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

07 Mar 2017, 13:29
As per question stem, number N has the three smallest prime numbers as it's only prime factors which means
N = 2^a * 3^b * 5^c

Question asks: How many positive integers divide n completely= which means, the total number of factors which N has.
This can be denoted as = (a+1)*(b+1)*(c+1)
So now we need to find out which statement(s) gives us the value of a, b & c.

Statement 1 - The total number of times the prime factors of n occur in n is 5.
This implies that, a+b+c = 5
Now we know (a,b,c) can take values such as (1,1,3) or (2,1,2)
In the two cases, total number of factors is different.
(1,1,3) => 2*2*4 = 16
(2,1,2) => 3*2*3 = 18

As we do not have a definite values, hence Statement 1 is not sufficient.

Statement 2 - The product of the number of times each prime factor of n occurs in n is 4.
This implies => a*b*c = 4
Now we know (a,b,c) can take values such as (1,1,4) or (2,1,2)
In the two cases, total number of factors is different.
(1,1,4) => 2*2*5 = 20
(2,1,2) => 3*2*3 = 18

As we do not have a definite values, hence Statement 2 is not sufficient.

Combining both Statements, we know

a+b+c =5 & a*b*c = 4
This means, that (a,b,c) can take any value (1,2,2) or (2,1,2) or (2,2,1), but the total number of factors will always remain as 18.

Therefore combining both Statement is sufficient.

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1755
Re: A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

27 Mar 2017, 05:41

Solution

Steps 1 & 2: Understand Question and Draw Inferences

The question tells us about a positive integer n which has smallest 3 prime numbers as its only prime factors. The questions asks us to find the number of positive integers that divide n completely
• Let’s first analyze the information given in the question statement. The prime factors of n are the 3 smallest prime numbers i.e. 2, 3 and 5.
• So, we can write $$n = 2^a * 3^b * 5^c$$ where a, b, c are positive integers
• We are asked to find the number of integers that divide n completely i.e. we are asked to find the number of factors of n.
• To find the number of factors of n, we need to know the following:
a. Number of prime factors of n à this information is known to us
b. Number of times the prime factors of n occur in nà this information is unknown to us.
• Number of factors of n = (a+1)(b+1)(c+1)
• So, we need to find the information about the number of times a prime factor of n occurs in n to find out the number of factors of n i.e. we need to find the values of a, b and c which give us a unique value of (a+1)(b+1)(c+1).
• With the above understanding, let’s see the information provided in the statements.

Step 3: Analyze Statement 1 independently

1. The total number of times the prime factors of n occur in n is 5
That is, a + b + c = 5.
As a, b, c are positive integers, their minimum values can be 1. Thus the possible values of {a, b, c} in any order can be:
• Case 1 - {2, 2, 1}
o In this case, Total number of factors = (2+1)*(2+1)*(1+1) = 3*3*2 = 18
• Case 2 - {3, 1, 1}
o In this case, Total number of factors = (3+1)*(1+1)*(1+1) = 4*2*2 = 16

As we do not have a unique answer, statement-1 is insufficient to answer the question.

Step 4: Analyze Statement 2 independently

2. The product of the number of times each prime factor of n occurs in n is 4.
That is a*b*c = 4.
As a, b, c are positive integers, their minimum values can be 1. Thus the possible values of {a, b, c} in any order can be:
• Case 1 - {4, 1, 1}
o In this case, Total number of factors = (4+1)*(1+1)*(1+1) = 5*2*2 = 20
• Case 2 - {2, 2, 1}
o In this case, Total number of factors = (2+1)*(2+1)*(1+1) = 3*3*2 = 18
As we do not have a unique answer, statement-2 is insufficient to answer the question

Step 5: Analyze Both Statements Together (if needed)

1. From Statement 1, we inferred that {a, b, c} = {2, 2, 1} or {3, 1, 1} in any order
2. From Statement 2, we inferred that {a, b, c} = {4, 1, 1} or {2, 2, 1} in any order

The only set of values that satisfy both the statements is {a, b, c} = {2, 2, 1} in any order.
Therefore, we can now find a unique value of product of a+1, b+1 and c+1

Thanks,
Saquib
Quant Expert
e-GMAT

Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

_________________

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Senior Manager
Joined: 06 Jul 2016
Posts: 417
Location: Singapore
Concentration: Strategy, Finance
A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

18 Aug 2017, 02:34
1
EgmatQuantExpert wrote:
Q. A positive integer n has the smallest 3 prime numbers as its only prime factors. How many positive integers divide n completely?

n = $$2^a$$*$$3^b$$*$$5^c$$.
(a+1)(b+1)(c+1) = ??
We need the values for a, b, c.

Quote:
(1) The total number of times the prime factors of n occur in n is 5.
(2) The product of the number of times each prime factor of n occurs in n is 4.

Let's solve Statement 2 first, as Statement 1 was convoluted to begin with, for me at least. I had to look at Statement 2 to see that Statement 1 meant Sum.

2) a*b*c = 4
a=1, b=2, c=2
a=4, b=1, c=1
Insufficient.

1) S2 had product, so this statement will have the sum
a + b + c = 5
Insufficient.

1+2)
a*b*c=4
a+b+c=5
=> the values have to be 2, 2, 1
=> (a+1)(b+1)(c+1) = 3*3*2 = 18
Sufficient.

_________________

Put in the work, and that dream score is yours!

Intern
Joined: 18 Aug 2015
Posts: 17
Location: United States
GPA: 3.38
Re: A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

18 Aug 2017, 11:18
I did not think 1 was a prime number...
Math Expert
Joined: 02 Sep 2009
Posts: 47037
Re: A positive integer n has the smallest 3 prime number [#permalink]

### Show Tags

18 Aug 2017, 11:21
mcm2112 wrote:
I did not think 1 was a prime number...

1 is NOT a prime number.

A Prime number is a positive integer with exactly two distinct divisors: 1 and itself. So, the smallest prime (and the only even prime) is 2.
_________________
Re: A positive integer n has the smallest 3 prime number   [#permalink] 18 Aug 2017, 11:21
Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.