AMC 10 · 2021 · #18

Grade 7 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

prime-factorizationfunction-evaluationfraction-arithmeticprime-numbers convert-to-algebraguess-and-check ↑ Prerequisites: prime-factorization

📏 Long solution 💡 2 insights

Problem

A function on the positive rationals turns products into sums: the value at a product is the sum of the values. It also gives every prime its own value. Among the five fractions listed, find the one whose value is negative.

Pick an answer.

(A)

$\frac{17}{32}$

(B)

$\frac{11}{16}$

(C)

$\frac79$

(D)

$\frac76$

(E)

$\frac{25}{11}$

How to solve

Strategy Convert to Algebra

Tool #13 (Algebra) — the functional rule f(ab) = f(a) + f(b) together with f(p) = p fully determines f on positive rationals: factor each number into primes, sum up primes for the numerator, subtract the prime-sum for the denominator. Once we have this clean formula f((Π p_i^a_i)/(Π q_j^b_j)) = Σ a_i p_i - Σ b_j q_j, Tool #6 (Guess & Check) plugs each of the five choices in and Tool #3 (Eliminate) discards every positive value. This problem is multiple choice with only five candidates — checking them all is the right move.

1STEP 1

Derive the basic properties

A reciprocal flips the sign.

f(1) = 0, f (1/x) = -f(x)

2STEP 2

Write the value of any fraction

Numerator primes add; denominator primes subtract.

f (N/D) = Σ a_i p_i - Σ b_j q_j

3STEP 3

Test the first choice

The first comes out positive.

f (17/32) = 17 - 10 = 7 ＞ 0

4STEP 4

Test the second choice

The second is positive too.

f (11/16) = 11 - 8 = 3 ＞ 0

5STEP 5

Test the third choice

The third is also positive.

f (7/9) = 7 - 6 = 1 ＞ 0

6STEP 6

Test the fourth choice

The fourth is positive as well.

f (7/6) = 7 - (2 + 3) = 2 ＞ 0

7STEP 7

Test the last choice

The negative one is twenty-five elevenths.

f (25/11) = 10 - 11 = -1 ＜ 0 → (E)

Answer

25/11

The big idea — f acts like a weighted prime counter — predicts when f(N/D) ＜ 0: the prime sum of D must exceed the prime sum of N. Only (E) flips that comparison: N = 25 = 5 · 5 contributes 5 + 5 = 10, while D = 11 contributes 11, and 11 ＞ 10. The other four choices have a single large prime upstairs (17, 11, 7, 7) and small repeated primes downstairs, so the upstairs total wins.

💡Key takeaway

f acts like a recipe: chop the number into its prime ingredients, add up the primes upstairs, subtract the primes downstairs. The only choice where the downstairs prime 11 beats the upstairs primes 5+5=10 is 25/11, giving f = -1 — so the answer is (E).