Competition · AMC preparation · step 4 of 4
AMC 10 · 2002B · #11
Grade 8 arithmeticPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Naming the middle integer n (Tool #4) is the smart move because the three consecutive numbers become n-1, n, n+1 — a symmetric setup that makes both the product and the sum collapse into clean expressions. Tool #13 (Convert to Algebra) then turns the sentence 'the product is 8 times the sum' into one equation in n. That equation shrinks to n²=25, and Tool #3 (Eliminate Possibilities) uses the word 'positive' to keep n=5 and discard n=-5.
Name the middle number
Call the middle integer n; being consecutive, the other two are n-1 and n+1.
For evenly spaced numbers, naming the middle one lets the outer two balance around it.
6.EE.B.6Introduce A VariableWrite the product and the sum
The ends cancel, so the sum is 3n; the outer pair folds, so the product is n(n²-1)=n³-n.
The symmetric n-1 and n+1 make the sum's ends cancel and the product fold into a difference of squares.
The symmetric outer terms make the sum's ends cancel and the product fold into a difference of squares.
▸ Why?
Terms placed symmetrically about the middle always pair to the same total, namely twice the middle.
▸ Why?
A number one more times a number one less is a difference of two squares, which collapses at once.
Solve for the middle number
The condition reads n³-n=24n, so n(n²-25)=0; n is positive, leaving n²=25 and n=5.
Dividing out the shared factor n collapses a cubic into n²=25, and 'positive' picks the single valid root.
8.EE.A.2Eliminate PossibilitiesAdd up the squares
So the integers are 4,5,6 and 4²+5²+6²=16+25+36=77, choice (B).
Once the numbers are pinned down, the answer is just three squares added together.
6.EE.A.1Convert To AlgebraFor three consecutive numbers, name the middle one n: the sum becomes 3n and the product becomes n³-n, and the equation shrinks to n²=25.
- Name the middle number
- Write the product and the sum
- Solve for the middle number
- Add up the squares
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