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## #1 2023-11-10 23:03:32

paulb203
Member
Registered: 2023-02-24
Posts: 250

### Factors. Can negative numbers be factors?

A question on Maths Genie;

Write down all the factors of 14.

Which is correct

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## #2 2023-11-10 23:15:11

Jai Ganesh
Registered: 2005-06-28
Posts: 47,391

### Re: Factors. Can negative numbers be factors?

Hi paul203,

With reference to factors, factors are whole numbers/natural numbers.

Negative integers are not used for factors of whole numbers.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #3 2023-11-10 23:55:52

paulb203
Member
Registered: 2023-02-24
Posts: 250

### Re: Factors. Can negative numbers be factors?

Thanks, Jai Ganesh

Interesting. Because the first half dozen or so Google results I got said that factors can be negative.

It does seem right that they would count as factors. We're told that a factor is a number that when multiplied by certain other numbers gives us a particular product (2, when multiplied by 9, gives us the product 18). -2 and -9 seem to fit this criteria.

But I trust your good self, and this site generally, so I'm now doubting those Google results, and my own intution.

Who, or what body, is the ulitmate arbiter of these things? Is there an ultimate arbiter of these things?

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## #4 2023-11-11 00:12:12

Jai Ganesh
Registered: 2005-06-28
Posts: 47,391

### Re: Factors. Can negative numbers be factors?

Hi,

Negative numbers factor similarly to positive numbers but the process is slightly different. A negative number will always have one negative factor. The number of factors for a negative number will therefore be double the number of factors for its positive counterpart.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #5 2023-11-11 00:39:15

paulb203
Member
Registered: 2023-02-24
Posts: 250

### Re: Factors. Can negative numbers be factors?

Thanks.

But I'm not talking about finding the factors of a negative number. I'm talking about finding the factors of a positive number. And wondering whether negatives can be included in the answer. There doesn't seem to me to be a consensus regards the answer.

“A negative number will always have one negative factor. The number of factors for a negative number will therefore be double the number of factors for its positive counterpart.”

1.    Is this implicitly telling us that factors CAN be negative? Or is it implicitly telling us that factors can be negative BUT ONLY FOR A NEGATIVE NUMBER?
2.     Take -9. -9 would then have the factors, 1 and -9, -1 and 9, 3 and -3. Which would be three negative factors, not one as stated in the Google result.

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## #6 2023-11-11 01:17:23

Bob
Registered: 2010-06-20
Posts: 10,514

### Re: Factors. Can negative numbers be factors?

There are a number of sites I refer to when I'm in doubt.

Wolfram Alpha gives the factors of 9 as {1, 3, 9}

Mathwords.com says "Factor of an Integer: Any integer which divides evenly into a given integer.
For example, 8 is a factor of 24.

That would include negatives.

MIF gives the factors of 12 as 1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, -12

In the UK National Curriculum factors are introduced in year 5; negative numbers in year 6. There is no mention of negatives being factors.

The BBC Bitesize website for helping students with GCSE maths gives this example:

"The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24."

You seem to have hit on an area where there are two different opinions about this.  In the other thread I have said that it is allowable to use your own definitions subject to certain rules.

If I was preparing a teenager for GCSE maths I would stick with positives only.  Other curricula may differ.

In doing questions involving Highest Common Factor
only positives are considered and so also with Prime Factorisation

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #7 2023-11-12 11:49:51

amnkb
Member
Registered: 2023-09-19
Posts: 253

### Re: Factors. Can negative numbers be factors?

paulb203 wrote:

Interesting. Because the first half dozen or so Google results I got said that factors can be negative.

It does seem right that they would count as factors. We're told that a factor is a number that when multiplied by certain other numbers gives us a particular product (2, when multiplied by 9, gives us the product 18). -2 and -9 seem to fit this criteria.

But I trust your good self, and this site generally, so I'm now doubting those Google results, and my own intution.

Who, or what body, is the ulitmate arbiter of these things? Is there an ultimate arbiter of these things?

i think the need for 'unique prime factorization' means factors should all have the same sign
otherwise lots of math gets messed up?

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## #8 2023-11-17 23:00:56

paulb203
Member
Registered: 2023-02-24
Posts: 250

### Re: Factors. Can negative numbers be factors?

Thanks, everyone.

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