2021 AMC 12A Problem 18

Problem 18 of 25IntermediateAlgebra Number Theory

Let

f

be a function defined on the set of positive rational numbers with the property that

f(a\cdot b) = f(a) + f(b)

for all positive rational numbers

a

and

b.

Suppose that

f

also has the property that

f(p) = p

for every prime number

p.

For which of the following numbers

x

f(x) \lt 0?

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Solution

The functional equation makes

f

completely additive: for

x = \prod p^{e_p},

we have

f(x) = \sum e_p\, f(p) = \sum e_p\, p,

where a prime in the denominator contributes a negative exponent (since

f(\frac{1}{p}) = -p

). Evaluating:

f\!\left(\tfrac{17}{32}\right) = 17 - 5\cdot 2 = 7,

f\!\left(\tfrac{11}{16}\right) = 11 - 4\cdot 2 = 3,

f\!\left(\tfrac{7}{9}\right) = 7 - 2\cdot 3 = 1,

f\!\left(\tfrac{7}{6}\right) = 7 - 2 - 3 = 2,

and

f\!\left(\tfrac{25}{11}\right) = 2\cdot 5 - 11 = -1.

Only the last is negative. Thus, the correct answer is E.

Tagged: functional equation · prime factorization