AMC 10 · 2021 · #12
Grade 6 number-theoryPick an answer.
Listing all 18 divisors of 768 and all 16 divisors of 384 by hand and adding them is slow and error-prone. The shortcut is to rewrite each number in prime-power form. Both are a power of 2 times 3, so every divisor is 2^a or 2^a × 3, and the whole divisor sum factors into (a chunk of powers of 2) times (1 + 3). That splits one long addition into two short ones, and the powers of 2 add to a number that is one less than the next power of 2. After that only a subtraction of two fractions remains. A second pass, viewing f(n) as a sum of reciprocals of divisors, checks the result from a completely different angle.
Rewrite both numbers with primes
Rewrite both with primes.
Prime-power form turns a big number into a short recipe, and here the two recipes differ by a single factor of 2.
6.EE.A.1Organize Information In More WaysDescribe every divisor at once
Describe every divisor at once.
Choosing a divisor is just choosing how many 2s and whether to include the 3, so the divisors organize themselves into a grid.
Choosing a divisor is just choosing how many twos and whether to include the odd prime.
▸ Why?
Every number has exactly one prime recipe, so a divisor is fully described by its exponents.
▸ Why?
Those choices are made independently, so the divisors organize themselves into a tidy grid.
Factor the divisor sum into a product
The divisor sum factors into a product.
The 3-row is just the 2-row scaled by 3, so the long sum collapses into one product of two short sums.
6.EE.A.3Identify SubproblemsBuild the two fractions
Build the two fractions.
Matching the denominators before reducing all the way makes the comparison a one-line subtraction.
4.NF.A.1Identify SubproblemsSubtract the matched fractions
With matched denominators the gap is tiny.
Once denominators agree, subtracting fractions is nothing but subtracting the top numbers.
5.NF.A.1Identify SubproblemsCheck it with reciprocals of divisors
Checking with reciprocals confirms one over one hundred ninety-two.
Dividing by n flips every divisor into a reciprocal, so the difference is only the reciprocals of the divisors 768 has that 384 lacks.
6.EE.A.3Change Focus Count The ComplementWrite the number as a power of 2 times 3, and its divisor sum factors into two short sums; dividing by the number turns divisors into reciprocals, so 768 beats 384 by only the reciprocals of the two divisors it gains.
- Rewrite both numbers with primes
- Describe every divisor at once
- Factor the divisor sum into a product
- Build the two fractions
- Subtract the matched fractions
- Check it with reciprocals of divisors