GMAT Functions: The function f is defined for each positive…

The function f is defined for each positive three-digit integer n by f (n) = 2^{x} \times 3^{y} \times 5^{z}, where x, y, and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f (m) = 9 f (v), then m - v =

functionsexponentsprime factorizationequations of primes

Explanation	Calculations	Help
$We let the digits of m be x_{m}, y_{m}, and z_{m}, and the digits of v be x_{v}, y_{v}, and z_{v} .$	$f (m) = 2^{x_{m}} \times 3^{y_{m}} \times 5^{z_{m}} f (v) = 2^{x_{v}} \times 3^{y_{v}} \times 5^{z_{v}}$
$We express the ratio of f (m) to f (v) in terms of the digit differences.$	$\frac{f ( m )}{f ( v )} = 2^{x_{m} - x_{v}} \times 3^{y_{m} - y_{v}} \times 5^{z_{m} - z_{v}}$
$We set this ratio equal to 9 and write 9 as a power of primes.$	$\frac{f ( m )}{f ( v )} = 3^{2}$
$We match each prime exponent on both sides to find how the digits differ.$	$x_{m} - x_{v} = 0 y_{m} - y_{v} = 2 z_{m} - z_{v} = 0$
$We express m - v in terms of the digit differences.$	$m - v = 100 \times (x_{m} - x_{v}) + 10 \times (y_{m} - y_{v}) + (z_{m} - z_{v})$
$We substitute the differences and calculate.$	$m - v = 100 \times 0 + 10 \times 2 + 0 = 20$

20 (D)

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