AMC 10 · 2022 · #3
Grade 5 number-theoryPick an answer.
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
The first term is a square.
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
The second also splits into a product.
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
The third fits the same shape.
Three matching cases is enough to trust the rule and write it in general form.
4.OA.C.5Look For A PatternWrite the general rule
Every term factors in the same shape.
Multiplying by 10ⁿ shifts a number n places left, which is what the splitting did each time.
5.NBT.A.2Look For A PatternCheck both factors
Both factors exceed one.
If both factors are bigger than 1, the product has divisors other than 1 and itself.
If both factors are bigger than one, the number has divisors other than one and itself.
▸ Why?
Every number has exactly one prime recipe, so a genuine factorization settles the question for good.
▸ Why?
Divisors come in pairs that multiply back to the number, so each factor names its own partner.
Count the primes
There are 0 primes.
A factorization with two factors ＞ 1 rules out primality, by definition.
4.OA.B.4Look For A PatternThis AMC 12 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.