Finding Factor Pairs for Whole NumbersDate: 11/29/2005 at 16:48:09 From: Tansu Subject: What is a factor pair I know you have already answered this question, but I still need help. What are the factor pairs of 66? Every time I try I don't really understand. Date: 11/29/2005 at 17:12:15 From: Doctor Ian Subject: Re: What is a factor pair Hi Tansu, Let's look at a smaller number, like 18. If we want to find all the factor pairs, we can try dividing 18 by every number up to 18, and see which ones divide it evenly: 18 / 1 = 18 18 / 2 = 9 18 / 3 = 6 18 / 4 = 18 / 5 = 18 / 6 = 3 18 / 7 = 18 / 8 = 18 / 9 = 2 18 / 10 = 18 / 11 = 18 / 12 = 18 / 13 = 18 / 14 = 18 / 15 = 18 / 16 = 18 / 17 = 18 / 18 = 1 Now, we can notice a couple of things. The first is that it's pointless to try any numbers greater than half of 18, because there's no way they can divide evenly. (Do you see why?) Also, a number is always divisible by itself. So we could have done this: 18 / 1 = 18 18 / 2 = 9 18 / 3 = 6 18 / 4 = 18 / 5 = 18 / 6 = 3 18 / 7 = 18 / 8 = 18 / 9 = 2 18 / 18 = 1 The second thing to notice is that each of these divisions corresponds to a multiplication. That is, 18 / 1 = 18 -> 18 = 18 x 1 18 / 2 = 9 18 = 9 x 2 18 / 3 = 6 18 = 6 x 3 18 / 4 = 18 / 5 = 18 / 6 = 3 18 = 3 x 6 18 / 7 = 18 / 8 = 18 / 9 = 2 18 = 2 x 9 18 / 18 = 1 18 = 1 x 18 So these are the factor pairs of 18: 18 and 1 9 and 2 6 and 3 3 and 6 2 and 9 1 and 18 Does this make sense? At this point, a second thing we can notice is that the later part of the list is the same as the earlier part, but with the factors in the opposite order: 18 and 1 -------- 9 and 2 ----- | 6 and 3 -- | | | | | These are the same, with order reversed 3 and 6 -- | | 2 and 9 ----- | 1 and 18 -------- So really, we can stop dividing when the quotient is smaller than the number we're dividing by: 18 / 1 = 18 18 / 2 = 9 18 / 3 = 6 18 / 4 = 4 remainder 2 18 / 5 = 3 remainder 3 <-- We can quit here There's no reason to continue after this point, because any pair we would get, we'll already have found. In general, the stopping point is reached when we get to the square root of the number whose factor pairs we're finding. For 18, the square is somewhere between 4 (4^2 = 16) and 5 (5^2 = 25), so 5 is the upper bound of the numbers we want to check. For 66, the square would be between 8 (8^2 = 64) and 9 (9^2 = 81). So although 66 seems like a much bigger number than 18, there aren't a lot more numbers to check: 66 / 1 = 66 / 2 = 66 / 3 = 66 / 4 = 66 / 5 = 66 / 6 = 66 / 7 = 66 / 8 = 66 / 9 = Does this make sense? Let me know if this helps... or if it doesn't. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 11/29/2005 at 18:30:50 From: Tansu Subject: Thank you (What is a factor pair) Dear Dr. Ian Thank you so much for replying! It means a lot to me, it really does. Everytime I try to write to someone else they NEVER reply. Now I know I can rely on you guys.....and understand things when I need help! Thanks again! - Tansu (aka Tunz) |
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