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What four, positive, dissimilar numbers have a sum that equals its product?

psycoswimmer

psycoswimmer

macrumors 65816
Original poster
Sep 27, 2006
1,302
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USA
All in the title really.

My little sister's math teacher gave them this as a "fun" problem, and I can't get it. I tried google but the answers there didn't help me much.

Basically, what I need is something like "1+1+1+1=1*1*1*4" but following the right criteria.

Any math geniuses to the rescue? :D
 
I was thinking the same thing, Abstract. :D

Thanks for that siurpeeman, but I need 4, not 3. :eek:
 
psycoswimmer said:
All in the title really.

My little sister's math teacher gave them this as a "fun" problem, and I can't get it. I tried google but the answers there didn't help me much.

Basically, what I need is something like "1+1+1+1=1*1*1*4" but following the right criteria.

Any math geniuses to the rescue? :D
Click to expand...

Do they have to be integers? Or any positive numbers?

For example, 1, 2, 3, 6/5 works

If you allow repeats then the integers 1,1,2,4 work
 
psycoswimmer said:
I was thinking the same thing, Abstract. :D

Thanks for that siurpeeman, but I need 4, not 3. :eek:
Click to expand...

oops, i misread that. math is my thing, not english. :)

1*(1/4)*(19/4)*32 = 38

1+(1/4)+(19/4)+32 = 38

further, you could make up your own numbers for three of them and solve for the last.

example (we'll use swiftaw's solution):

1 + 2 + 3 + x = 1*2*3*x

6 + x = 6x

6 = 5x

x = 6/5 or 1.2
 
siurpeeman said:
oops, i misread that. math is my thing, not english. :)

1*(1/4)*(19/4)*32 = 38

1+(1/4)+(19/4)+32 = 38

further, you could make up your own numbers for three of them and solve for the last.

example (we'll use swiftaw's solution):

1 + 2 + 3 + x = 1*2*3*x

6 + x = 6x

6 = 5x

x = 6/5 or 1.2
Click to expand...

Exactly, there are an infinite number of solutions if you are dealing with the positive real numbers.

However, if you restrict to positive integers only, I believe that the only solution is 1,1,2,4, although I don't have a rigorous proof to support that claim.

Edit: I now have a mathematical proof that 1,1,2,4 is the only set of 4 positive integers for which their sum and product are equal.
 
How about: -2, 5, -3, 0.

It works...

edit: forgot they had to be positive.
 
There are many answers to this question, since this is a quad algebraic equation with only 2 known conditions. Answers are infinite and rather simple, although positive integer answers are kinda tough.

One possible answer is 0.89 (8/9), 1, 2 and 5.
 
To summarize what I said above.

4 Positive real numbers: Infinite correct answers
4 Positive Integers: 1 correct answer = 1,1,2,4
 
swiftaw said:
There are no set of 4 different positive integers for which sum = product.
Click to expand...

I am only stating the title of this thread, which if I'm not wrong is the question of this problem. You need 4 different positive numbers.

If we take the question as it's phrased, then these numbers don't have to be integers, and I think it's safe to think so considering it's a problem for young children.
 
I guess, to me, the question as phrased makes very little sense because if it is just find 4 unique positive numbers that have sum = product, then that is trivial as there are an infinite number of solutions.

If the numbers have to be positive integers, then we also have a problem because there is no set of 4 positive integers that has sum = product and has all 4 numbers being different because the ONLY set of 4 positive integers that has sum = product is 1,1,2,4.
 
Thanks for the input since I left.

Swiftaw provided me with both a correct answer (which I was looking for), and, on a separate note, the only way to answer it with integers.

As it turns out, my sister told me the wrong information the first time around and her teacher said that there could be a repeat of one number. In this case, swiftaw's "1,1,2,4" answer is the correct answer.

I'm still having her use a fraction one though to be different. :D
 
This question is purposely phrased in this way to test the analytical and reasoning abilities of children to solve this problem. It stands to reason that since this is a quad (4 unknown) algebraic equation, by assuming any 3 values to any 3 of the unknowns, the 4th can be found, hence the infinite possibilities. The path towards this analytical process is the reason for such an abstract yet meaningful problem. It's the process that the teacher is interested in teaching, not solving the problem.
 
If you didn't use variables to solve this problem, how would go about solving it?

My sister went to the same school I went to, had the same math teacher, etc. and I know that they don't learn variables until next year. So the "1+2+3+x=1*2*3*x" method wouldn't go over well with them. So I'm guessing that she just wanted them to do guess and check? I don't see how that's helping them analyze and reason as orpheus1120 suggested.
 
You could start trying 1 x 2 x 3 x 4, and note that the product is 24 and the sum is 10. So it has to involve smaller numbers, which means something has to be repeated. And soon you'll see 1 1 2 4 is it, as noted above.
 
Hls811 said:
1,1,2,4

this was in this past Sundays Parade Magazine
Click to expand...

Ahh, this could be where she got it from.

I kind of like these problems though. My school might be starting a half-credit class for these types of problems and I might sign up.
 
I still don't see how 1,1,2,4 is the answer if they are supposed to be dissimilar.
 
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