Positive real numbers a and b have the property that √\log a + √\log b + \log √a + \log √b = 100 and all four terms on the left are positive integers, where \log denotes the base 10 logarithm. What is ab?

2019 AMC 12A Problem 15 — Solution

2019 AMC 12A Problem 15

Below is the professionally curated solution for Problem 15 of the 2019 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12A solutions, or check the answer key.

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Concepts:logarithm Diophantine Equation casework

Difficulty rating: 1730

15.

Positive real numbers $a$ and $b$ have the property that

$\sqrt{\log a} + \sqrt{\log b} + \log \sqrt{a} + \log \sqrt{b} = 100$

and all four terms on the left are positive integers, where $\log$ denotes the base $10$ logarithm. What is $ab?$

$10^{52}$

$10^{100}$

$10^{144}$

$10^{164}$

$10^{200}$

Solution:

Let $\sqrt{\log a} = p$ and $\sqrt{\log b} = q,$ so $\log a = p^2$ and $\log\sqrt{a} = \dfrac{p^2}{2}.$ For this to be an integer, $p$ is even; likewise $q.$

Writing $p = 2m,$ $q = 2n,$ the equation $p + q + \dfrac{p^2}{2} + \dfrac{q^2}{2} = 100$ becomes $m(m+1) + n(n+1) = 50.$

The only solution is $\{m, n\} = \{4, 5\},$ giving $\log(ab) = p^2 + q^2 = 4(16 + 25) = 164.$

Therefore $ab = 10^{164}.$

Thus, the correct answer is D.

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