AMC 10 · 2011 · #4
Grade 4 number-theoryPick an answer.
The wrong product 161 is the end of Ron's mistake, and we need the numbers he started from, so Tool #11 (Work Backwards) fits: factor 161 to recover the two numbers he actually multiplied. Tool #3 (Eliminate Possibilities) then picks which factor is the reversed two-digit number — only one factor has two digits. Tool #7 (Identify Subproblems) breaks the rest into small steps: un-flip the digits to get a, then do the correct multiplication.
Factor the wrong product
The wrong product splits into two primes.
Splitting the wrong answer into factors reveals the two numbers Ron actually multiplied.
Splitting the wrong answer into factors reveals the two numbers that were actually multiplied.
▸ Why?
Every number has exactly one prime recipe, so the ways to write it as a product are fixed in advance.
▸ Why?
Divisors come in pairs that multiply back to the number, so naming one factor names its partner.
Pick the reversed two-digit factor
Only one of them has two digits.
A two-digit number can only be a two-digit factor, which eliminates every other choice.
1.NBT.B.2Eliminate PossibilitiesUn-flip the digits to get a
Flipping its digits back recovers the true factor.
Reversing the reversal returns the original number.
1.NBT.B.2Identify SubproblemsMultiply correctly
Multiplying gives 224, choice (E).
Once the true digits are back in place, the correct product is a plain multiplication.
4.NBT.B.5Identify SubproblemsBreak the wrong answer into its factors, flip the two-digit one back to the real number, then multiply again to get the true product.
- Factor the wrong product
- Pick the reversed two-digit factor
- Un-flip the digits to get a
- Multiply correctly