AMC 8 · 2000 · #14
Grade 5 arithmeticnumber-theoryPick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We never need the full 19¹⁹ or 99⁹⁹ — only their last digit. Tool #9 (Solve an Easier Related Problem) collapses both giants to the same easier question: "what is the units digit of 9ⁿ?" since both bases end in 9. Tool #5 (Look for a Pattern) then takes over: compute 9¹, 9², 9³, 9⁴ by hand, spot the length-2 cycle 9, 1, 9, 1, …, and read off the units digit for any exponent. Add the two units digits at the end and take the units digit of that small sum.
Only the last digit matters, so each base collapses to its units digit 9: units(19¹⁹) = units(9¹⁹), units(99⁹⁹) = units(9⁹⁹).
Long multiplication shows the ones-place digit of a product only ever depends on the ones-place digits of the factors — a Grade 5 multi-digit multiplication fact.
5.NBT.B.5Solve An Easier Related ProblemList the first few powers of 9 and read off the units digits, stopping as soon as the pattern repeats.
Hand-multiplying four small powers is well within Grade 5 fluency and exposes the cycle without algebra.
5.NBT.B.5Look For A PatternThe units digits cycle 9, 1, 9, 1 — length 2: an odd exponent gives 9, an even one gives 1.
Naming a repeating rule from a short list is the Grade 4 "generate a number pattern" move.
4.OA.C.5Look For A PatternBoth exponents 19 and 99 are odd, so units(9¹⁹) = units(9⁹⁹) = 9.
Once the cycle is known, every exponent question becomes a parity check — odd or even.
4.OA.C.5Look For A PatternAdd the two units digits: 9 + 9 = 18, whose units digit is 8.
The units digit of a sum depends only on the units digits of the addends — a Grade 4 place-value-aware addition.
4.OA.A.3Look For A PatternOnly the last digit matters: powers of 9 flip between 9 and 1, both exponents are odd so each piece ends in 9, and 9 + 9 = 18 ends in 8.