AMC 10 · 2014 · #7
Grade 7 number-theoryPick an answer.
Plugging in values of n one at a time finds examples but never proves the list is finished, so the plan has two jobs: bound the search, then make the test exact. Tool #14 (Extreme Principle) supplies the bound — the denominator must stay positive, which traps n between 1 and 29. Tool #16 (Change Focus) does the real work: stop watching n and watch the complement d=30-n instead, because n/(30-n)=30/d-1 turns the whole question into a single divisibility test on d. Tool #4 names that complement, and Tool #2 lists the divisors of 30 from its prime factorization, which guarantees the list is complete rather than merely long.
Bound the search range
Positivity bounds the search.
A positive divided by a negative is negative, so the bottom has to stay on the positive side.
7.NS.A.2Extreme PrincipleName the complement d
Naming the complement simplifies the fraction.
The denominator is doing all the work, so give it a name and let n ride along behind it.
6.EE.B.6Introduce A VariableRewrite the fraction using d
It becomes a constant over that complement.
Splitting the fraction peels off a constant and leaves exactly one division to worry about.
6.EE.A.3Change Focus Count The ComplementTurn it into a divisibility test
So the condition is a divisor test.
"Comes out whole after dividing" is just the word divisor, so the condition becomes a factor list.
Coming out whole after dividing is exactly what the word divisor means, so the condition becomes a factor list.
▸ Why?
A quotient is a whole number exactly when the division leaves no remainder.
▸ Why?
Every number has one prime recipe, so its divisors can be listed completely with nothing missed.
List the divisors and count
Counting divisors gives 7, choice (D).
The prime factorization certifies that the divisor list is complete, so counting it is counting the answers.
4.OA.B.4Make A Systematic ListWhen a fraction has to come out whole, rewrite it so the bottom must divide one fixed number — then the answer is just a divisor count.
- Bound the search range
- Name the complement d
- Rewrite the fraction using d
- Turn it into a divisibility test
- List the divisors and count