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# How many positive three digit integers have exactly 7 factor

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Re: How many positive three digit integers have exactly 7 factor [#permalink]
v1rok wrote:
(A) None.

This was actually pretty easy once I realized that exactly 7 factors can happen only if these seven factors are prime numbers.

So I started writing down 7 smallest prime numbers:

2,3,5,7,11,13,19

If you start multiplying them the product pretty quickly becomes larger than 3-digit number.

You are missing 1 and the no. itself.. so,for that we need to consider 5 prime #s by ur approach, which yields the same answer though..
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
Hint: What is the smallest positive integer with EXACTLY 7 factors?
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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kevincan wrote:
Hint: What is the smallest positive integer with EXACTLY 7 factors?

1*2*3*5*7*11*13 =30030 , isn't it ?
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
Another clue: If an integer has exactly 7 factors, how many prime factors must it have?

15=5*3 has 4 factors , 5^0*3^0 , 5^0*3^1, 5^1*3^0 and 5^1,5^1
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
Right! 15 has 4 factors but only 2 prime factors.
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
kevincan wrote:
Right! 15 has 4 factors but only 2 prime factors.

Anyone has a say on this ? I am not sure what's correct.. but somehow used to thinking that 7 factors mean minimum 7 prime factors where they don't repeat..
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
1000 has two prime factors, but 16 factors!
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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kevincan wrote:
How many positive three digit integers have exactly 7 factors?

(A) none (B) one (C) two (D) three (E) more than three

I think I found one... but this question took me way tooooooo much time. I just could not forget about this problem and it was bothering me...

I think the question is unique in the sense that it wants exactly 7 factors. Which means that it has to be some form of a perfect square.
ie. 13*13 = 169, this has 3 factors = 1, 13, 169

So my first guess was 25*25 = 625, but I was only able to find the following factors; 1,5,25,125,625.

but how about 27*27 = 729
I was able to find 7 factors for 729
1x729
3x249
9x81
27x27

therefore: 1, 3, 9, 27, 81, 249, 729

in advance, if I missed a factor in 729...or if I totally mis-understood the question disregard this message...

kevin, any more help?
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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If p1 and p2 are two different prime numbers, how many factors does
p1*p2 have? 4=2*2

and p1^n1*p2^n2 if n1 and n2 are postive integers? (n1+1)(n2+1)

So, if an integer has seven (a prime number) factors, it must be that...
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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7 factors, means the number is a perfect square, also means that 7 is a prime so our number have power like (x+1) = 7
so number has power of 6

2pow6= 64 not 3 digits
3pow6= 27 x 27 = cool
5pow6 = not 3 digit integer

so only one number

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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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Property of prime nos- Any prime no. has only 2 factors-1 and itself.
Eg. For X-a prime no., factors are 1 & x
When you multiply the prime no with itself i.e, square the number, it has three factors
x^2--factors are 1, x, x^2-3 factors.
When you cube it, it has 4 factors-1,x^, x^2 & x^3.
for x^n, we have n+1 factors.

In the question, n+1=7=> n=6. and the requite no ishould be in the form x^6.
Now try with the first prime-2
2^6=64- Not a 3 digit no.
3^6=729-YES. It's a 3 digit no.
5^6=625*625 Big no....No need to check further.
Hence, there is one and only one three-digit no (729) which has 7 factors.
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
I used logic to find this answer.

Prob: Need all 3 digit numbers that exactly have 7 factors.
Soln: For any number to have an odd number of factors, the number has to be a perfect square. Hence I limited my set starting from 10^2 to 31^2 i.e. {10^2, 11^2, 12^2, 13^2.......... 31^2}.
- Why? Because I need to evaluate 3 digit numbers that are perfect squares

Step1. You can safely remove all the prime numbers from the above set ( {Set} - {11,13,17,19,23,29,31})
---since all prime numbers would exactly have 3 factors eg: 11^2 --> 1, 11, 121

Step2. You can safely remove all numbers from the above set which have 2 or more prime factors ({Set} - {10,12,14,15,16,18,20,21,22,24,26,28,30})
---since all numbers having 2 or more prime factors would lead to the 3 digit number having more than 7 factors
eg: 22^2--> 1, 2,4,11,22,44,121,242,484.
eg: 15^2--> 1, 3,5,9,15,25,45,75,225.

Step3. You are left with 25, 27 -
25 ^2- 5 factors
27^2- 7 factors
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
kevincan wrote:
How many positive three digit integers have exactly 7 factors?

(A) none
(B) one
(C) two
(D) three
(E) more than three

key point to know is that only square of prime numbers have odd number of factors
in this case (x^2)^3 ; x^6 ; so x has to be prime number which would be 3 as only 3^6 is a 3 digit number ...
option B
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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Re: How many positive three digit integers have exactly 7 factor [#permalink]
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