For any positive integer n, n1, the "length" of n is the number of po

Question

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessarily distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5.

Which of the following integers has length 3 ?

A) 3

B) 15

C) 60

D) 64

E) 105

sashiim20 · Accepted Answer

"Length" of n is the number of positive primes (not necessarily distinct).

Check all the options.
A) 3 = 3 x 1
Number of positive primes = 1

B) 15 = 3 x 5
Number of positive primes = 2

C) 60 = 2 x 2 x 3 x 5
Number of positive primes = 4

D) 64 = 2 x 2 x 2 x 2 x 2 x 2 =  2^6 
Number of positive primes = 6

E) 105 = 3 x 5 x 7 
Number of positive primes = 3
Therefore integer which has length 3 = 105.
Answer E...

Bunuel · Answer

Questions about the same concept (the length of an integer): https://gmatclub.com/forum/for-any-posi ... 90320.html https://gmatclub.com/forum/for-any-posi ... 26368.html https://gmatclub.com/forum/the-length-o ... 88734.html https://gmatclub.com/forum/the-length-o ... 32624.html https://gmatclub.com/forum/for-any-posi ... 40950.html https://gmatclub.com/forum/for-any-inte ... 08124.html https://gmatclub.com/forum/what-is-the- ... 32111.html Hope it helps.

AliciaSierra · Answer

105=5*7*3. So length is 3

utkarshthapak · Answer

I believe the point we have to remember is that this is not a direct question on no. of factors but an indirect one. It can be solved by factorizing the given no. For e.g., 60 = 2X2X3X5 So its length is 4. Hope it helps someone

JeffTargetTestPrep · Answer

Let’s test each answer choice:
 
A) 3
 
Since 3 is a prime number, its length is 1. 
 
B) 15
 
5 x 3 = 15, so 15 has a length of 2.
 
C) 60
 
60 = 10 x 6 = 2 x 5 x 3 x 2, so 60 has a length of 4.
 
D) 64 
 
64 = 2 x 2 x 2 x 2 x 2 x 2, so 64 has a length of 6.
 
E) 105
 
105 = 7 x 5 x 3, so 105 has a length of 3.
 
Answer: E

SrinidhiIlan · Answer

Why can't 15 be written as 5 x 3 x 1?

Bunuel · Answer

Check the question:

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessarily distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5.

Which of the following integers has length 3 ?

A) 3

B) 15

C) 60

D) 64

E) 105

1 is  not  a prime number.

egmat · Answer

Hi SrinidhiIlan,

Good question, and I can see exactly where this is coming from. You're looking at sashiim20's list where  15 = 3 × 5  gets a length of  2 , and you're asking: if I toss in a  1 , doesn't that make it  5 × 3 × 1 , so length  3 ?

The catch is in the definition itself.  The "length" counts  positive primes  whose product is n. And  1 is not a prime number.  A prime has exactly two factors (1 and itself);  1  has only one factor, so it's excluded by rule. That means  1  can never be one of the factors you're allowed to count - no matter how many times you write it.

So for  15 , the only prime factorization is  3 × 5 . That's two primes, length  2 . Tacking on ×  1  doesn't add a prime; it just multiplies by  1 , which changes nothing.

Here's why the rule has to work this way:  if  1  were allowed to count, then  every  number would have unlimited length. Watch:

-  15 = 3 × 5 × 1  - "length  3 "
-  15 = 3 × 5 × 1 × 1  - "length  4 "
-  15 = 3 × 5 × 1 × 1 × 1  - "length  5 "

The "length" would never settle on one value - the definition would be useless. Keeping it to  primes only  makes each number's length a single, fixed number.

Therefore ,  105 = 3 × 5 × 7  is the only choice with length exactly  3 . Every factor there is a genuine prime, and there are exactly three of them.

Answer: E

For any positive integer n, n>1, the "length" of n is the number of po

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