Competition · AMC preparation · step 4 of 4
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.
Keep only the units digits
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 powers of 9
List 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 PatternSpot the cycle
The 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.
For a whole number whose ones digit is 9, the ones digit of its power follows a fixed two-step cycle: an odd exponent leaves ones digit 9, and an even exponent leaves ones digit 1.
▸ Why?
Each next power is the previous power multiplied by 9, and the ones digit of that product is decided only by the ones digit of the previous power, so the ones digits form a sequence you can follow on their own, ignoring every higher digit.
▸ Why?
Split the number being multiplied into its ones digit plus its tens-and-higher part; only the (ones digit times 9) piece can reach the ones place of the answer.
▸ Why?
Breaking one factor into a sum and multiplying each part by 9 separately is exactly the rule that lets a product spread across a sum.
▸ Why?
The tens-and-higher part times 9 is still a whole number of tens, and a whole number of tens has 0 in the ones place, so it cannot change the ones digit of the total.
▸ Why?
Tracking just that ones digit, 9 times a number ending in 9 ends in 1, and 9 times a number ending in 1 ends in 9, so the ones digit flips 9, 1, 9, 1 and comes back to its start after every two multiplications.
▸ Why?
A step that always returns to the same value after the same fixed number of moves is a repeating cycle, so exponents two apart share a ones digit and the length-2 pattern fixes every case by whether the exponent is odd or even.
Apply the rule to both
Both 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 and take the units digit
Add 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.
- Keep only the units digits
- List the powers of 9
- Spot the cycle
- Apply the rule to both
- Add and take the units digit
A parent dashboard for the family lives at sensimlab.com.