AMC 8 · 2002 · #4
Grade 4 number-theoryPick an answer.
AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
A 4-digit palindrome always has the shape ABBA — the thousands digit matches the units digit, and the hundreds digit matches the tens digit. Tool #5 (Look for a Pattern) captures that shape. Once we lock the shape in, Tool #6 (Guess and Check) finishes the job: keep A = 2 to stay close to 2002, and try the next value of B.
A four-digit palindrome must read ABBA — the thousands digit matches the units, the hundreds the tens.
Each digit gets a place value; the palindrome rule pins the units digit to the thousands digit and the tens digit to the hundreds digit.
4.NBT.A.2Look For A PatternFit 2002 to ABBA: A = 2, B = 0, and the tens digit 0 and units digit 2 confirm the pattern.
Reading 2002 as ABBA shows the starting values we need to bump.
4.NBT.A.2Look For A PatternKeep A = 2 and raise B from 0 to 1 (making A = 3 would jump all the way to 3003), which gives 2112.
Increasing B by 1 raises the year by 110 (B in the hundreds and B in the tens), which is the smallest possible jump while A stays at 2.
4.NBT.A.2Guess And CheckMultiply the four digits of 2112.
Multiplying by 1 leaves the value unchanged, so the product is just 2 × 2.
3.OA.C.7Look For A PatternA 4-digit palindrome always reads ABBA, so the next one after 2002 just needs the smallest bump in the middle digit — that gives 2112, and its digits multiply to 4.