Competition · AMC preparation · step 4 of 4
AMC 8 · 2015 · #14
Grade 4 algebranumber-theoryPick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We do not yet know which sums are possible, so start by listing a few real sums of four consecutive odd integers (Tool #9): 1+3+5+7, 3+5+7+9, 5+7+9+11, and so on. Then look at the resulting sums for a pattern (Tool #5) — every sum jumps by 8, so every possible sum is a multiple of 8. Finally test the five answer choices against that pattern (Tool #3, Eliminate) — only one choice fails the divisibility-by-8 test.
Try a few small sums
Start easy: add the four smallest consecutive odd integers, then shift up, listing the totals 16, 24, 32, 40.
Doing several small additions by hand is a Grade 4 multi-digit addition skill, and it gives us real data to look at.
4.NBT.B.4Solve An Easier Related ProblemFind the jump between sums
Each total is 8 more than the last, so every possible sum is a multiple of 8.
Spotting a constant jump of 8 in a list of numbers and using that to describe every term is exactly the Grade 4 "generate and analyze patterns" idea. (Why 8? Each odd integer goes up by 2, and there are 4 of them, so the sum goes up by 4 × 2 = 8.)
The sum of any four consecutive odd integers is always a multiple of 8.
▸ Why?
The smallest such group, 1+3+5+7, adds to 16, and 16 is exactly 2 groups of 8, so the very first sum is already a multiple of 8.
▸ Why?
Stepping from one group of four consecutive odd integers to the next raises the sum by exactly 8, and adding one more 8 to a total that was already a stack of 8s leaves it a stack of 8s.
▸ Why?
Each of the four numbers goes up by 2, and four steps of 2 make 4 × 2 = 8, so the whole sum goes up by 8.
▸ Why?
A sum equal to 8 × k becomes 8 × (k+1) after another 8 is added, since 8 × k + 8 = 8 × (k+1) pulls out the shared factor 8.
Test each choice for a multiple of 8
Test each choice: the one that is not a multiple of 8 is the impossible sum.
Checking each choice against the rule "must be a multiple of 8" is the Grade 4 multiples-and-factors test, and on a multiple-choice problem it is the fastest way to finish.
4.OA.B.4Eliminate PossibilitiesPick the one that fails
Only 100 is not a multiple of 8, so it is the number that cannot be written this way — the answer is (D).
Exactly one choice survives the elimination, which matches the problem's promise of one impossible value.
4.OA.B.4Eliminate PossibilitiesAdding four small odd numbers a few times and spotting the jump-by-8 pattern is a Grade 4 skill — no algebra needed to crack this AMC 8 problem!
- Try a few small sums
- Find the jump between sums
- Test each choice for a multiple of 8
- Pick the one that fails
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