Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #17
Grade 4 arithmeticnumber-theoryPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The sum has 1011 terms and the last term 2022!! is astronomically large, so direct calculation is hopeless. Tool #9 (Easier Related Problem) says: compute just the first few double factorials and watch what happens to their units digits. Tool #5 (Look for a Pattern) then takes over: once we see that 10!! ends in 0, the factor 10 inside every later n!! (n ≥ 10) guarantees a units digit of 0 for every later term, so only the first four terms (2!!, 4!!, 6!!, 8!!) can influence the answer. After that the problem collapses to a one-line addition.
Compute the first few terms
Build the first few double factorials by multiplying one more even number each time: 2, 8, 48, 384.
Multiplying a small multi-digit number like 48 by a one-digit number like 8 is exactly the Grade 4 multi-digit multiplication skill.
4.NBT.B.5Solve An Easier Related ProblemList the units digits
List the units digits 2, 8, 8, 4, then compute 10!! = 384 · 10 = 3840 — it ends in 0.
Spotting that multiplying by 10 tacks a 0 onto the end is the Grade 3 "identify arithmetic patterns" skill in action.
3.OA.D.9Look For A PatternSpot the factor of 10
Every even n ≥ 10 has 10 as a factor, so 10!! through 2022!! all end in 0.
Recognizing the rule "any product with a factor of 10 ends in 0" is a Grade 3 multiplication-pattern observation, and it makes thousands of terms drop out at once.
For every even integer n ≥ 10, the double factorial n!! = 2 · 4 · 6 · 8 · 10 … n ends in the digit 0.
▸ Why?
The value n!! is some whole number multiplied by 10, and any whole number times 10 ends in 0.
▸ Why?
Once n reaches 10, the list of factors 2 · 4 · 6 · 8 · 10 … n contains 10, so n!! can be written as (the product of the other even factors) × 10.
▸ Why?
Rearranging and regrouping the factors so the × 10 is done last gives the exact same product, so pulling the 10 out to the end is allowed.
▸ Why?
Multiplying a whole number by 10 shifts every digit up one place and leaves a 0 in the ones place, so the result ends in 0.
Add the four that matter
Adding 0 changes nothing, so only 2!! + 4!! + 6!! + 8!! count: 2 + 8 + 8 + 4 = 22, whose units digit gives the result.
Adding four single-digit numbers and reading the ones place is a Grade 2 fluency skill — the hard work was all in the pattern, not the arithmetic.
2.NBT.B.5Look For A PatternThis AMC 8 problem only needs Grade 4 multiplication and one pattern you already know — anything times 10 ends in 0!
- Compute the first few terms
- List the units digits
- Spot the factor of 10
- Add the four that matter
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