For any positive integer n, the length of n is defined as th

Question

For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.

A. None 
B. One 
C. Two 
D. Three 
E. Four

Bunuel · Accepted 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.

Bunuel · Answer

For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.

A. None 
B. One 
C. Two 
D. Three 
E. Four

Basically the length of the integer is the sum of the powers of its prime factors.

Length of six means that the sum of the powers of primes of the integer (two digit) must be  6 . First we can conclude that  5  can not be a factor of this integer as  the smallest integer with the length of six  that has  5  as prime factor is  2^5*5=160  (length=5+1=6), not a two digit integer.

The above means that the primes of the two digit integers we are looking for can be only  2  and/or  3 .  n=2^p*3^q ,  p+q=6  max value of  p  and  q  is  6 .

Let's start with the highest value of  p :
 n=2^6*3^0=64  (length=6+0=6);
 n=2^5*3^1=96  (length=5+1=6);

n=2^4*3^2=144  (length=4+2=6) not good as 144 is a three digit integer.

With this approach we see that actually  5<=p<=6 .

Answer: C.

Hope it helps.

mbaiseasy · Answer

My Solution:

Try increasing prime numbers with length 6:
Trial 1:  2^6=64  Valid
Trial 2:  3^6 =729  Invalid

This means our candidate 2-digit numbers have combinations of  2  and  3

2^6=64 
 2^5x3^1=96 
 2^4x3^2=144  Invalid

Answer: Two

YTT · Answer

That's brilliant!!! I especially love the part where I could take 5 away. This really save tons of time! Thanks!! BTW, Thanks so much for the prompt response!

mmphf · Answer

The best (quickest) way I can think of to get the answer, is start with 2^6, then move on from there.

2^6=64
2^5*3=96

Obviously 2^5*5 will be more than 2 digits, as will 2^4*3^2. So 64 and 96 are it. Answer is 2 (C).

You may have been looking for something even faster, but this is fast enough for me. Unless someone has a better way.

Spunkerspawn · Answer

Try the smallest possible value first: In this case it is 2^6 which equals 64.

If we replace the last 2 with 3, then we have 2^5*3 = 96

From here we can positively assume that any other number will have more than 2 digits. So the answer is (C) 2 numbers that have length 6 and are only 2 digits.

Daddydekker · Answer

Just writing it out took me .46 sec.:

length of 6, lets take the lowest prime factor 2

2x2x2x2x2x2 = 64

Now substitute the last 2 by a 3, and see the solution gets 96. We can think of what will happen when we substitute another 2 for a three.

Hence, C (2)

OptimusPrepJanielle · Answer

For the length to be 6, the number of prime factors should be maximum. Hence we need to use maximum 2's

The numbers can be 2^6 = 64 and 2^5*3 = 96 
For any other number less than 100, the length will be less than 6

Correct option: C

JeffTargetTestPrep · Answer

We need to determine how many 2-digit integers have a length of 6, or in other words how many 2-digit integers are made up of 6 prime factors. Let’s start with the smallest possible numbers:
 
2^6 = 64 (has a length of 6)
 
2^5 x 3^1 = 96 (has a length of 6)
 
Since 2^4 x 3^2 = 144 and 2^5 x 5^1 = 160 are greater than 99, there are no more 2-digit numbers that have a length of 6.
 
Answer: C

abhishekdadarwal2009 · Answer

For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.

A. 0 
B. 1 
 C. 2 
D. 3 
E. 4

So it is evident that the length can only include 2 and 3 also when we try with 2 we find that length longer than 6 is not possible to be a 2 digit number.

and only one 2 can be replaced by 3 and we get 96,

this can be solved by hit and trial by starting from the lower prime and then moving up.

64 and 96 are the only two numbers possible to have length 6.

matthewsmith_89 · Answer

Bunuel 
You made this question look easy. Thanks for your explanation.

sashiim20 · Answer

Lets start with smallest prime number  2 .

2^6 = 64  ---------- (Length  = 6 )

2^7  is three digit number hence cannot be  n .

Therefore lets move to next prime number  3 .

2^5*3^1 = 32*3 = 96  ---------- (Length  = 6 )

2^4*3^2  will be three digit number, hence cannot be  n .

Therefore we have   "Two"   two-digit positive integers which have length  6   = 64  and  96

Answer (C)...

vivapopo · Answer

2^6=64 2^5*3=96 2^4*3^2=144 out So ans.C

dave13 · Answer

Bunuel but 2^6 is already 64 and if we multiply it by 3 we get 192 n=2^6*3^0=64 how can it be equal 64

(length=6+0=6);

Bunuel · Answer

We are not multiplying it by 3, we are multiplying by 3^0, which is 1.

BrentGMATPrepNow · Answer

Let's first find the smallest value with length 6. 
This is the case when each prime factor is 2. 
We get 2 x 2 x 2 x 2 x 2 x 2 = 64. This is a 2-digit positive integer. PERFECT

To find the next largest number with length 6, we'll replace one 2 with a 3
We get   3   x 2 x 2 x 2 x 2 x 2 = 96. This is a 2-digit positive integer. PERFECT

To find the third largest number with length 6, we'll replace another 2 with a 3 
We get  3  x   3   x 2 x 2 x 2 x 2 = 144. This is a 3-digit positive integer. NO GOOD

So there are only  two  two-digit positive integers with length 6.

Answer: C

Cheers,
Brent

prateekchugh · Answer

For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.

According to the question, 
n=  a^p*b^q*c^r 
Length = p+q+r

We need to find two-digit positive integers whose length=6; means, p+q=6

Case 1)  2^6 ; length=6; n=64
Case 2)  2^5 * 3^1 ; length=6; n=96
Case 3)  2^4 * 3^2 ; length=6; n=144 Three-Digit Incorrect
Case 5)  2^3 * 3^3 ; length=6; n=216 Three- Digit Incorrect
As we increase, the value of n will increase beyond 2-digit positive integer

A. None
B. One
C.  Two     Answer  
D. Three
E. Four

egmat · Answer

Hey there! I can see you're working through this tricky number properties question. Let me walk you through the key insights that'll help you crack this one.

Understanding the Definition 
First, let's make sure we're crystal clear on what "length" means here. The length of a number is simply counting  all  prime factors in its prime factorization, including repeats.

For example:  75 = 3 	imes 5 	imes 5 
When we count all prime factors: 3, 5, 5 → that's 3 factors total
So the length of 75 is 3. ✓

Finding Two-Digit Numbers with Length 6 
Now we need two-digit numbers (10-99) that have exactly 6 prime factors when we count them all.

Let's think systematically. To get 6 prime factors while keeping our number small, we should use the smallest possible primes.

Using Powers of 2 
The smallest prime is 2, so let's see what we get:
 
  2^6 = 64  (length = 6) ✓ This works! It's two-digit. 
  2^7 = 128  (length = 7) - This is three digits, so too big.

Using Mixed Primes  
What about combinations like  2^5 	imes 3 = 32 	imes 3 = 96 ?
This also has length 6 (five 2's and one 3) and is two-digit! ✓

Let's check:  2^4 	imes 3^2 = 16 	imes 9 = 144 
This is three digits, so it doesn't work.

What about  2^3 	imes 3^3 = 8 	imes 27 = 216 ? 
Also three digits.

Systematic Check 
So we found:
 
  64 = 2^6  (length 6) ✓ 
  96 = 2^5 	imes 3  (length 6) ✓

Notice how any other combination either gives us a three-digit number or doesn't achieve length 6 in the two-digit range.

Answer: C. Two

The complete systematic approach for identifying  all possible combinations  and the underlying patterns that help you solve similar problems faster can be found in the  detailed solution on Neuron . You'll also discover the strategic framework that applies to all "constrained optimization" problems in number properties. For more practice with similar official questions and comprehensive explanations, check out  Neuron's complete question bank .

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