34 split so sum=product

Probability Level 4

Divide the set of integers from 1 to 34 into two subsets A & B where the sum of the numbers in set A is equal to the product of the numbers in set B.

Enter this sum = product.

The answer is 544.

2 solutions

Jeremy Galvagni
May 10, 2018

If n is even then B can consist of the numbers $1$ , $(\frac{n}{2})-1$ , $n$ . The product of these is $\frac{n^{2}}{2}-n$

The sum of the numbers in A is then $\frac{n^{2}+n}{2}-1-(\frac{n}{2}-1)-n=\frac{n^{2}}{2}-n$

So B can consist of $1$ , $16$ , $34$ with a product of $1*16*34=\boxed{544}$

Note: for most n, there is more than one solution, but for n=34 the solution is unique. As far as I can tell it is the largest number with a unique solution.

How do you know that set B has exactly 3 elements?

Pi Han Goh - 3 years, 1 month ago

I didn't, but I found the formula for a set of three numbers that works for any even n. (Showing the n=34 solution is unique was harder.) There's a similar formula for odd n.

I just found small solutions and looked for a pattern.

Jeremy Galvagni - 3 years ago

Giorgos K.
May 11, 2018

I made a program in $Mathematica$ testing all triplets as product and got the right answer quickly

Select[Subsets[Range@34,{3}],Times@@#==Total@Complement[Range@34,#]&]

this returns {1,16,34} in a sec

34 split so sum=product

The answer is 544.

2 solutions

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