AMC 10 · 2015 · #10
Grade 7 algebranumber-theoryPick an answer.
As written, x+y+xy=80 is a sum, and a sum of 80 can be split in a huge number of ways — there is nothing to grab. Tool #15 (Organize Information in More Ways) rewrites the same equation as a single product, because products of integers are rigid: a fixed product can only be built from that number's divisors. Tool #2 (Make a Systematic List) then lays out every divisor pair, and Tool #3 (Eliminate Possibilities) uses x ＞ y ＞ 0 to delete all of them but one. The whole solution turns on one thing worth being careful about: the divisor argument is only legal because both factors are proved to be positive integers.
Add 1 to force a product
Adding one makes the left side a product.
A sum of 80 can happen a thousand ways, but a product of 81 can happen almost no ways — so trade the sum for a product.
Adding one to each side turns a sum with a thousand possibilities into a product with almost none.
▸ Why?
Expanding the shifted product reproduces the original sum plus the constant, so the two forms are the same statement.
▸ Why?
A product of two whole numbers can only split the ways its divisors allow, and those come in a short list.
Check the factors are positive integers
Both factors are whole and at least two.
"Product equals 81, so the parts are factors of 81" is only true for whole-number factors, so the whole-number part has to be earned first.
6.EE.B.8Eliminate PossibilitiesList every divisor pair of 81
There are only three divisor pairs.
A number that is a single prime raised to a power has a short, complete divisor list you can write down with no risk of missing one.
4.OA.B.4Make A Systematic ListUse x ＞ y ＞ 0 to delete pairs
The ordering leaves exactly one.
The bigger factor of a pair must beat the square root of the product, so x+1 has to exceed 9.
7.EE.B.4Eliminate PossibilitiesRead off and verify x
Reading it off gives 26, choice (E).
A solution is not finished until the numbers are put back into the original equation and the inequalities are re-checked.
6.EE.B.5Eliminate PossibilitiesWhen an equation mixes a sum and a product of two unknowns, add 1 to both sides and see whether it collapses into a single product — a product of positive integers can only happen in a handful of ways.
- Add 1 to force a product
- Check the factors are positive integers
- List every divisor pair of 81
- Use x ＞ y ＞ 0 to delete pairs
- Read off and verify x