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The function f is defined for each positive three-digit integer n by f (n) = 7^{a} \times 2^{b} \times 3^{c}, where a, b, and c are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f (m) = 14 \times f (v), what is the value of m - v ?

Explanation	Calculations	Help
$We first name the hundreds, tens, and units digits of m as a_{m}, b_{m}, and c_{m}, and the hundreds, tens, and units digits of v as a_{v}, b_{v}, and c_{v} .$	$f (m) = 7^{a_{m}} \times 2^{b_{m}} \times 3^{c_{m}} f (v) = 7^{a_{v}} \times 2^{b_{v}} \times 3^{c_{v}}$
$We then write the ratio of f (m) over f (v) in terms of the digit differences.$	$\frac{f ( m )}{f ( v )} = 7^{a_{m} - a_{v}} \times 2^{b_{m} - b_{v}} \times 3^{c_{m} - c_{v}}$
$Next, we set this ratio equal to 14 and break 14 into its prime factors.$	$\frac{f ( m )}{f ( v )} = 14 = 2 \times 7$
$We can recognise that this equation has unknowns in exponents, so we apply the Prime Matching rule to match the exponents of each prime factor on both sides.$	$a_{m} - a_{v} = 1 b_{m} - b_{v} = 1 c_{m} - c_{v} = 0$	Theory & Tactics Method Card ALG7 Click to view full details
$We express the difference m minus v in terms of these digit differences.$	$m - v = 100 \times (a_{m} - a_{v}) + 10 \times (b_{m} - b_{v}) + (c_{m} - c_{v})$
$We substitute the digit differences and find the value of the difference.$	$m - v = 100 \times 1 + 10 \times 1 + 0 m - v = 110$

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C

The function f is defined for each positive three-digit integer n by f (n) = 7^{a} \times 2^{b} \times 3^{c}, where a, b, and c are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f (m) = 14 \times f (v), what is the value of m - v ?

Explanation	Calculations	Help
$We first name the hundreds, tens, and units digits of m as a_{m}, b_{m}, and c_{m}, and the hundreds, tens, and units digits of v as a_{v}, b_{v}, and c_{v} .$	$f (m) = 7^{a_{m}} \times 2^{b_{m}} \times 3^{c_{m}} f (v) = 7^{a_{v}} \times 2^{b_{v}} \times 3^{c_{v}}$
$We then write the ratio of f (m) over f (v) in terms of the digit differences.$	$\frac{f ( m )}{f ( v )} = 7^{a_{m} - a_{v}} \times 2^{b_{m} - b_{v}} \times 3^{c_{m} - c_{v}}$
$Next, we set this ratio equal to 14 and break 14 into its prime factors.$	$\frac{f ( m )}{f ( v )} = 14 = 2 \times 7$
$We can recognise that this equation has unknowns in exponents, so we apply the Prime Matching rule to match the exponents of each prime factor on both sides.$	$a_{m} - a_{v} = 1 b_{m} - b_{v} = 1 c_{m} - c_{v} = 0$	Theory & Tactics Method Card ALG7 Click to view full details
$We express the difference m minus v in terms of these digit differences.$	$m - v = 100 \times (a_{m} - a_{v}) + 10 \times (b_{m} - b_{v}) + (c_{m} - c_{v})$
$We substitute the digit differences and find the value of the difference.$	$m - v = 100 \times 1 + 10 \times 1 + 0 m - v = 110$

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C