A positive integer n has the smallest 3 prime number

Question

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.

Answer Choices   :

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.

Thanks,
Saquib
Quant Expert
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EgmatQuantExpert · Answer

The official solution has been posted. Looking forward to a healthy discussion..

blackfish · Answer

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.

Answer is C.

quantumliner · Answer

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.

Answer is C

EgmatQuantExpert · Answer

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

Step 3: Analyze Statement 1 independently

o In this case, Total number of factors = (2+1)*(2+1)*(1+1) = 3*3*2 = 18

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

o In this case, Total number of factors = (4+1)*(1+1)*(1+1) = 5*2*2 = 20

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)

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

Sufficient to answer

Answer: C

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

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akshayk · Answer

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

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.

C is the answer.

mcm2112 · Answer

I did not think 1 was a prime number...

Bunuel · Answer

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.

bumpbot · Answer

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A positive integer n has the smallest 3 prime number

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