AMC 10 · 2009 · #5
Grade 4 number-theoryPick an answer.
Name the twin age t and Kiana's age k. Since the twins share an age, the product becomes t × t × k = 128, so t shows up twice. That lets me list only the whole-number twin ages whose square divides 128, then throw out the ones that break the 'older than Kiana' rule.
Name the ages
The equal pair means the product holds a square.
Using one letter for the twins captures the fact that their two ages are really the same number.
4.OA.A.3Introduce A VariableList the twin ages that fit
Listing them gives four candidates.
Since the twin age is used twice, only ages whose square fits inside 128 are even possible.
Since the twins' age is used twice, only ages whose square fits inside the product are possible.
▸ Why?
The product has one prime recipe, so the three ages must be a genuine factoring of it.
▸ Why?
Any larger twin age would already push the product above the given value on its own.
Keep only older twins
The older condition leaves only one.
The 'older brothers' fact is a filter that removes every factor set except the one where the twins are bigger than Kiana.
4.NBT.A.2Eliminate PossibilitiesAdd the three ages
Adding gives 18, choice (D).
Once the only valid ages are found, adding them is the last easy step.
4.NBT.B.4Introduce A VariableWhen two people are the same age, that age gets used twice in the product, so only ages whose square fits the number can work.
- Name the ages
- List the twin ages that fit
- Keep only older twins
- Add the three ages