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The function f is defined for each positive three-digit integer n by f (n) = 3^{h} 2^{t} 7^{u}, where h, t, and u are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f (m) = 8 f (v), then m - v =

Explanation	Calculations	Help
$We name the digits of m as h_{m}, t_{m}, and u_{m}, and the digits of v as h_{v}, t_{v}, and u_{v} .$	$f (m) = 3^{h_{m}} \times 2^{t_{m}} \times 7^{u_{m}} f (v) = 3^{h_{v}} \times 2^{t_{v}} \times 7^{u_{v}}$
$We write the ratio of f (m) to f (v) in terms of the digit differences.$	$\frac{f ( m )}{f ( v )} = 3^{h_{m} - h_{v}} \times 2^{t_{m} - t_{v}} \times 7^{u_{m} - u_{v}}$
$We know this ratio equals eight, which is a power of two.$	$\frac{f ( m )}{f ( v )} = 2^{3}$
$We can recognise this because the equation has unknowns in exponents, so we apply Prime Matching to match the exponents of each prime.$	$h_{m} - h_{v} = 0 t_{m} - t_{v} = 3 u_{m} - u_{v} = 0$	Theory & Tactics Method Card ALG7 Click to view full details
$We express the difference m - v in terms of the digit differences.$	$m - v = 100 \times (h_{m} - h_{v}) + 10 \times (t_{m} - t_{v}) + (u_{m} - u_{v})$
$We substitute the values of the digit differences and then we find the result.$	$m - v = 100 \times 0 + 10 \times 3 + 0 = 30$

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C

The function f is defined for each positive three-digit integer n by f (n) = 3^{h} 2^{t} 7^{u}, where h, t, and u are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f (m) = 8 f (v), then m - v =

Explanation	Calculations	Help
$We name the digits of m as h_{m}, t_{m}, and u_{m}, and the digits of v as h_{v}, t_{v}, and u_{v} .$	$f (m) = 3^{h_{m}} \times 2^{t_{m}} \times 7^{u_{m}} f (v) = 3^{h_{v}} \times 2^{t_{v}} \times 7^{u_{v}}$
$We write the ratio of f (m) to f (v) in terms of the digit differences.$	$\frac{f ( m )}{f ( v )} = 3^{h_{m} - h_{v}} \times 2^{t_{m} - t_{v}} \times 7^{u_{m} - u_{v}}$
$We know this ratio equals eight, which is a power of two.$	$\frac{f ( m )}{f ( v )} = 2^{3}$
$We can recognise this because the equation has unknowns in exponents, so we apply Prime Matching to match the exponents of each prime.$	$h_{m} - h_{v} = 0 t_{m} - t_{v} = 3 u_{m} - u_{v} = 0$	Theory & Tactics Method Card ALG7 Click to view full details
$We express the difference m - v in terms of the digit differences.$	$m - v = 100 \times (h_{m} - h_{v}) + 10 \times (t_{m} - t_{v}) + (u_{m} - u_{v})$
$We substitute the values of the digit differences and then we find the result.$	$m - v = 100 \times 0 + 10 \times 3 + 0 = 30$

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C