AMC 10 · 2005 · #8
Grade 6 number-theoryPick an answer.
Searching all 1000 digit triples is hopeless by hand, but the equation is a product of two whole numbers, and 2005 has very few divisors. So list every divisor of 2005, use the largest possible digit values to cap each factor, and cross out every pairing that breaks a cap. If only one pairing survives, the digits are forced — which is stronger than merely exhibiting a triple that happens to work.
Name the two factors
Both factors are whole, so each is a divisor of the product.
Giving the two clumps names turns a digit puzzle into a plain question about factoring one number.
6.EE.B.6Introduce A VariableFactor 2005 completely
Factoring completely leaves only four divisors.
A number with only two prime factors has only four divisors, so the whole search space is four values wide.
Because 2005 is built from just two primes, it has only four divisors, so the whole search is four values wide.
▸ Why?
Every number has exactly one prime recipe, so its divisors are fixed the moment that recipe is written down.
▸ Why?
Divisors come in pairs that multiply back to the number, so naming one factor names the other.
Cap each factor using the digit limit
Maximum digits cap the two factors at 999 and 27.
Testing the most extreme digits gives a hard ceiling that rules out whole branches at once.
6.EE.B.8Extreme PrincipleCross out every impossible pairing
Only one pairing survives both caps.
Two short lists make only six combinations, and checking all six leaves exactly one survivor.
4.OA.B.4Eliminate PossibilitiesRead the digits back out of 401
Place value reads the digits back, so the leading digit is 4, choice (D).
Place value runs in both directions: once the number is known, its digits are known too.
4.NBT.A.2Work BackwardsWhen two whole numbers multiply to a fixed value, list that value's divisors first — then the size limits on each factor usually leave only one choice.
- Name the two factors
- Factor 2005 completely
- Cap each factor using the digit limit
- Cross out every impossible pairing
- Read the digits back out of 401