factoring problem...

• May 4th 2009, 04:46 AM
Not2l8
factoring problem...
when you look list the possible factors of some even numbers they look like this -

60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60)

ie. Sometimes 2 numbers next to each other (in bold) on the list multiplied equal the number.

414 (1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414)

80 (1, 2, 4, 5, 8, 10, 16, 20, 40, 80)

My question is - If I have an even number that I know will fulfill this criteria. (ie. 2 certain numbers next to each other on its factor list multiplied will equal the original number) Is there a formula for finding out what these 2 numbers will be without having to figure out the whole factor list and manually going through it?
• May 4th 2009, 05:39 AM
Not2l8
I've just been told that the square root of the original number will get you to somewhere inbetween the two factors I'm looking for, however for really big numbers this will still make finding the factors hard. Any other ideas?
• May 4th 2009, 06:43 AM
TheAbstractionist
Quote:

Originally Posted by Not2l8
I've just been told that the square root of the original number will get you to somewhere inbetween the two factors I'm looking for, however for really big numbers this will still make finding the factors hard. Any other ideas?

Hi Not2l8.

You are on the right track. If $n$ is not a perfect square, take the two factors of $n$ closest to $\sqrt n,$ one less and the other greater than $\sqrt n.$

However, if $n$ is a perfect square, then what you observe will never happen, since $\sqrt n$ itself will be in the list of factors.
• May 4th 2009, 07:44 AM
Media_Man
A clear pattern
Not2l8,

You have correctly listed the divisors of 414:
414 (1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414)

Now list them this way and perform these multiplications (Wink):

414
18 x 23 =
09 x 46 =
06 x 69 =
03 x 138 =
02 x 207 =
01 x 414 =

This trick will always work for every single number (see if you can prove why), with a slight twist on the square numbers:

400
20 x 20
16 x 25
10 x 40
08 x 50
05 x 80
04 x 100
02 x 200
01 x 400

That being, that the square root of n is left without a partner on the list, so it will be partnered with itself. As for your question of finding a magic formula that would give you this closest pair, f(414)=18, f(400)=20. I think the simplest answer is: no. (Angry)