Thread: Can the product and sum of a set of digits identify them uniquely?

Can two (or more) different sets of five single digit numbers have the same product and sum of all their digits at the same time?

ie.
1,4,8,4,3 (Product = 384, Sum = 20)
2,5,3,1,9 (Product = 270, Sum = 20)
The sums are the same but the products aren’t so these two sets don’t.

Furthermore, does the answer differ for other sized sets of single digits, can you find two different sets of any amount of digits that have the same product and sum?

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I just realised that it's possible with 3 or more digits and at least one as a 0. But if none of the numbers were 0?

1,1,2,3,4 product = 24 sum = 11
1,2,2,2,3 product = 24 sum = 10

1,1,1,4,4 product = 16 sum = 11
1,1,2,2,4 product = 16 sum = 10

1,2,3,5,7 product = 210 sum = 18
1,1,5,6,7 product = 210 sum = 20

Anyone know if this is even possible yet? I tend to think it's not.

tough one!

I tried algebra but it didn't change much the difficulty

excel helped check many numbers fast, but didn't get me a solution,

then I tried some common sense and started from the multipliers
3,3,2,2,2
which I broke into 6,6,2 and 9,2,2 and I had spare "1" digits to fill the gap in the addition

the solution I found:
{6,6,2,1,1} {9,2,2,2,1}

(edit: of course the sets of size 3 {6,6,1} {9,2,2} also work)

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Originally Posted by Manu

tough one!

I tried algebra but it didn't change much the difficulty

excel helped check many numbers fast, but didn't get me a solution,

then I tried some common sense and started from the multipliers
3,3,2,2,2
which I broke into 6,6,2 and 9,2,2 and I had spare "1" digits to fill the gap in the addition

the solution I found:
{6,6,2,1,1} {9,2,2,2,1}

(edit: of course the sets of size 3 {6,6,1} {9,2,2} also work)

Yuwie.com | invite friends. hang out. get paid.

Awesome!!
I guess I gave up too early. I had gotten it down to where I would find a difference of at least 1 in the sum, as you can see in my previous post.. but I gave up too soon, before finding out how to eliminate that difference of 1.