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What is the largest 3 digit number to have an odd number of factors?

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What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 09 Mar 2016, 11:55
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 09 Mar 2016, 20:37
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Bunuel wrote:
What is the largest 3 digit number to have an odd number of factors?

A. 625
B. 729
C. 841
D. 943
E. 961


Some info on ODD/EVEN number of FACTORS..
1) ODD FACTORS-
a) if there are odd factors of an integer, that integer will be a square..
b) If the integer has 3 factors only, it is square of a PRIME number..

2) EVEN FACTORS-
a) If an integer has even number of factors, it will not be a Square..
b) If an integer has 2 factors, its a PRIME number

Here we are looking for largest 3-digit number, which has odd factors..

so this number must be square and since it asks for the largest 3-digit number..
We can look for the largest number given-961 , which is 31^2..

Also we know 900=30^2 so answer has to be equal to or greater than 30..
check for 31^2..

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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 09 Mar 2016, 19:38
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It is basically asking what is the biggest 3 digit square number. Answer is 'E'

Tip- All squares have odd numbers of factors.
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 09 Mar 2016, 20:51
All squares have odd number of factors.
Highest square 31^2 =961
Answer E
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 14 Mar 2016, 04:36
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 05 Jan 2018, 09:08
Bunuel wrote:
What is the largest 3 digit number to have an odd number of factors?

A. 625
B. 729
C. 841
D. 943
E. 961


A number greater than 1 will have an odd number of factors only if it's a perfect square. The largest 3-digit perfect square is 31 x 31 = 961 (since 32 x 32 = 1,024).

Answer: E
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 03 May 2018, 09:23
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Bunuel wrote:
What is the largest 3 digit number to have an odd number of factors?

A. 625
B. 729
C. 841
D. 943
E. 961


IMPORTANT CONCEPT: All positive integers have a EVEN number of positive factors EXCEPT integers that are squares of integers
Squares of integers (e.g., 1, 4, 9, 16, 25, 36, etc) have an ODD number of positive factors.

So, the question is really asking us "What is the largest 3 digit number that is the SQUARE OF AN INTEGER?"

Let's find out.
30² = 900, so 900 will have an odd number of positive factors
31² = 961, so 961 will have an odd number of positive factors
32² = 1024, so 1024 will have an odd number of positive factors.

Of course 1024 is a FOUR-DIGIT number.
So, 961 must be the greatest 3-digit number with an ODD number of positive factors.

Answer: E

Cheers,
Brent
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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New post 14 Sep 2018, 23:51
1
Just a question: How is one suppose to mentally calculate/know for certain that the answer choices presented are all perfect squares?

Is it common to memorize Perfect Squares up to 30..etc?.. I feel this question would take me far too much time to assess whether each answer choice is a perfect square. The only one I know off the bat was 625..:/
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Re: What is the largest 3 digit number to have an odd number of factors? &nbs [#permalink] 14 Sep 2018, 23:51
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