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

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Math Expert
Joined: 02 Sep 2009
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What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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09 Mar 2016, 12:55
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What is the largest 3 digit number to have an odd number of factors?

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

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Posts: 8017
Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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09 Mar 2016, 21:37
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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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09 Mar 2016, 20:38
1
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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09 Mar 2016, 21:51
All squares have odd number of factors.
Highest square 31^2 =961
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Re: What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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14 Mar 2016, 05:36
Remember whenever the question says odd number odd number of factors we must conclude that its a perfect square.
Hence E is correct which is a perfect square of 31
hence 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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05 Jan 2018, 10:08
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).

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

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03 May 2018, 10:23
Top Contributor
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.

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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15 Sep 2018, 00:51
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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What is the largest 3 digit number to have an odd number of factors?  [#permalink]

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02 Aug 2019, 22:25
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

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

Number having odd number of factors = square of integer
$$961 = 31^2$$

IMO 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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10 Sep 2019, 12:26
stonecold wrote:
Remember whenever the question says odd number odd number of factors we must conclude that its a perfect square.
Hence E is correct which is a perfect square of 31
hence E

How can you infer that 961 is the square of 31?
Do you use prime factorization or what?
Re: What is the largest 3 digit number to have an odd number of factors?   [#permalink] 10 Sep 2019, 12:26
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