AMC 10 · 2017 · #12
Grade 4 number-theoryPick an answer.
The hidden engine of this problem is one translation: Horse k returns to the start every k minutes, so at time T it stands at the start exactly when k divides T. Tool #4 (Introduce a Variable) lets us name the meeting time T and restate the whole race as a divisibility condition. Once that is clear, "at least 5 horses at the start" becomes "T has at least 5 divisors among 1 through 10," and the smallest such T is easy to hunt for by Tool #6 (Guess and Check): test T=1,2,3,… in order and count qualifying divisors, using Tool #2 (Make a Systematic List) to keep the tally tidy. Because we only need 5 of the 10 horses, T will be far below the all-meet time S=2520, so checking small numbers by hand is quick.
Turn position into divisibility
Being at the start is plain divisibility.
A horse only meets the start line at whole numbers of its own laps, so its return times are exactly the multiples of k.
A horse only meets the start line at whole numbers of its own laps, so its return times are the multiples of its lap time.
▸ Why?
After a full lap everything is back where it began, so the pattern repeats with that period.
▸ Why?
A time counts only when the division by the lap length leaves no remainder.
Restate the goal
The goal becomes a divisor count.
Asking for 5 horses is asking T to carry 5 small divisors at once — a question purely about factors.
4.OA.A.3Introduce A VariableSearch upward and count divisors
Searching upward finds 12 first.
Climb the numbers one at a time; the first one rich enough in small factors to feed five horses is the winner.
4.OA.C.5Guess And CheckAdd the digits of T
Its digits add to 3, choice (B).
Once you have the time itself, the digit sum is just a quick final addition.
4.OA.A.3Make A Systematic ListA horse is back at the start whenever the clock hits a multiple of its number, so finding when 5 horses meet is just finding the smallest time with 5 small divisors — that is 12, and its digits add to 3.
- Turn position into divisibility
- Restate the goal
- Search upward and count divisors
- Add the digits of T