The function f is defined for each positive three-digit integer n by...

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The function f is defined for each positive three-digit integer n by f(n)=2^x 3^y 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=

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20 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 3^y_m 5^z_m f(v) = 2^x_v 3^y_v 5^z_v We express the ratio of f(m) to f(v) in terms of the digit differences. f(m)/f(v) = 2^x_m - x_v 3^y_m - y_v 5^z_m - z_v We set this ratio equal to 9 and write 9 as a power of primes. 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 (x_m - x_v) + 10 (y_m - y_v) + (z_m - z_v) We substitute the differences and calculate. m - v = 100 0 + 10 2 + 0 = 20

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

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