Competition · AMC preparation · step 4 of 4
AMC 10 · 2022B · #6
Grade 5 number-theoryPick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Computing a₁₀ digit-by-digit and primality-testing a 21-digit number is hopeless. Tool #9 (Easier Problem) says: try the small cases n=1, 2, 3 first and look for shared structure. Tool #5 (Pattern) then notices that each small case splits the same way — a string of ones, shifted, plus a string of ones — which is a clean common factor. Tool #7 (Subproblems) breaks the prime question into two: 'find a factorization' and 'check both factors are ＞ 1'. If we can show every term factors into two pieces both bigger than 1, no term is prime.
Factor the first term
Smallest case: a₁ = 121 = 110 + 11 = 11 · (10 + 1) = 11 · 11 — composite.
Finding a factor pair is exactly the Grade 4 way to show a number is not prime.
4.OA.B.4Solve An Easier Related ProblemFactor the second term
n=2: a₂ = 11211 = 11100 + 111 = 111 · (100 + 1) = 111 · 101 — same shape.
Splitting the digits at the 2 lines up with a clean multiplication by a power of 10.
4.NBT.B.5Solve An Easier Related ProblemFactor the third term
n=3: a₃ = 1112111 = 1111000 + 1111 = 1111 · (1000 + 1) = 1111 · 1001 — pattern holds.
Three matching cases is enough to trust the rule and write it in general form.
4.OA.C.5Look For A PatternDescribe the factoring pattern
In general a_n = R_n+1 · (10ⁿ + 1): a run of n+1 ones times a 1, then zeros, then a 1.
Multiplying by 10ⁿ shifts a number n places left, which is what the splitting did each time.
Multiplying by a power of ten shifts a number that many places to the left.
▸ Why?
A number is its digits weighted by their places, so each place is ten times the one to its right.
▸ Why?
An exponent counts how many times a factor is used, so it counts how many shifts happen.
Check both factors exceed 1
For every n from 1 to 10 both factors exceed 1: R_n+1 ≥ 11 and 10ⁿ + 1 ≥ 11.
If both factors are bigger than 1, the product has divisors other than 1 and itself.
5.OA.A.2Identify SubproblemsCount the primes
Every a_n (n = 1…10) is a product of two factors ＞ 1 → all composite, none prime. Answer (A).
A factorization with two factors ＞ 1 rules out primality, by definition.
4.OA.B.4Look For A PatternThis AMC 10 problem only needs Grade 5 place value and factor pairs you already know — split each number at the 2, peel out the common string of ones, and every single one of the ten terms is a product of two whole numbers bigger than 1, so none are prime.
- Factor the first term
- Factor the second term
- Factor the third term
- Describe the factoring pattern
- Check both factors exceed 1
- Count the primes
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