AMC 10 · 2023 · #8
Grade 4 arithmeticPick an answer.
AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Computing 2022²⁰²³ exactly is impossible by hand, so Tool #9 (Easier Problem) says: shrink the question. The units digit of 2022²⁰²³ is the same as the units digit of 2²⁰²³, and the units digit of 2023²⁰²² is the same as the units digit of 3²⁰²² — both follow because the units digit of a product depends only on the units digits of the factors. Tool #5 (Pattern) then takes over: list the first few units digits of powers of 2 and powers of 3, notice each cycles with period 4, and use 2023 mod 4 and 2022 mod 4 to land in the right slot. Tool #7 (Subproblems) keeps the two pieces clean — solve each units digit separately, then add and read the ones place. No algebra, no big arithmetic.
A power's last digit depends only on its base's last digit, so 2022²⁰²³ matches 2²⁰²³ and 2023²⁰²² matches 3²⁰²².
Stripping each base to its units digit is the easier-problem move — Grade 3 properties of multiplication on the ones place.
3.OA.B.5Solve An Easier Related ProblemList the units digit of 2^k for small k to spot a pattern.
Writing out a small table to see a repeating block — Grade 3 arithmetic pattern hunting.
3.OA.D.9Look For A PatternThe block 2, 4, 8, 6 repeats every 4, so 2023 mod 4 = 3 lands on the 3rd entry, 8.
One division-with-remainder picks the cycle slot — Grade 4 multi-digit division.
4.NBT.B.6Look For A PatternRepeat for powers of 3. List the units digits of 3^k for small k.
Same small-table technique catches the (3, 9, 7, 1) block — Grade 3 arithmetic pattern hunting.
3.OA.D.9Look For A PatternThe block 3, 9, 7, 1 also repeats every 4, so 2022 mod 4 = 2 lands on the 2nd entry, 9.
Same Grade 4 division-with-remainder trick to read the right slot.
4.NBT.B.6Look For A PatternAdd the two units digits: 8 + 9 = 17, whose ones digit is 7.
Adding two single-digit numbers and reading the ones place — Grade 1 single-digit sum.
1.NBT.C.4Identify SubproblemsThis AMC 10 problem only needs Grade 4 division-with-remainder and pattern-spotting you already know — the units digit of 2^k cycles through 2, 4, 8, 6 and the units digit of 3^k cycles through 3, 9, 7, 1, so 2023 mod 4 = 3 picks 8 and 2022 mod 4 = 2 picks 9, and 8 + 9 = 17 ends in 7.