AMC 10 · 2018 · #19
Grade 6 number-theoryPick an answer.
Tool #9 (Solve an Easier Related Problem): the whole problem is about what divides n, so factoring 323 and tracking which prime factors must also divide n is the engine. Tool #4 (Introduce a Variable): naming the next divisor d lets us argue about it before we know its value. Tool #7 (Identify Subproblems): split the work into 'what must d look like' and then 'which choice is smallest and actually achievable'. Tool #3 (Eliminate Possibilities): once we know d must be a multiple of 17 or 19, two of the answer choices fail that test and drop out immediately.
Name the next divisor and list the facts
Give the next divisor a name.
Giving the unknown divisor a name lets us reason about it before we pin down a number.
4.NBT.A.2Introduce A VariableFactor 323 to see what must divide n
323 is a product of two primes.
A divisor's prime factors are forced to divide the whole number too.
A divisor's prime factors are forced to divide the whole number too.
▸ Why?
Every number has exactly one prime recipe, so a divisor's primes are already in that recipe.
▸ Why?
A divisor divides with no remainder, so everything inside it divides with no remainder as well.
Show the next divisor must share a factor of 17 or 19
If coprime, the number exceeds four digits.
If the next divisor shared nothing with 323, the number would have to swallow both, ballooning past four digits.
6.NS.B.4Solve An Easier Related ProblemFind the smallest multiple of 17 or 19 above 323
Find the smallest such multiple above 323.
Only multiples of 17 or 19 survive, and the very next one after 323 is 340.
4.OA.B.4Eliminate PossibilitiesCheck that 340 is actually reachable
Checking it is reachable gives 340.
Building the smallest number that contains both divisors confirms no divisor sneaks in between them.
6.NS.B.4Identify SubproblemsSince 323=17×19, the next divisor has to share a 17 or a 19, and the first one past 323 is 17×20=340.
- Name the next divisor and list the facts
- Factor 323 to see what must divide n
- Show the next divisor must share a factor of 17 or 19
- Find the smallest multiple of 17 or 19 above 323
- Check that 340 is actually reachable