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CAPITULO XII

Mínimo Común Múltiplo

Ejercicio 116

Hallar el m.c.m de:

$\begin{matrix} 2 a, 4 x - 8 \\ 2 a \\ 4 x - 8 = 2^{2} (x - 2) \\ m . c . m = 2^{2} a (x - 2) \\ m . c . m = 4 a (x - 2) \end{matrix}$
$\begin{matrix} 3 b^{2}, a b - b^{2} \\ 3 b^{2} \\ a b - b^{2} = b (a - b) \\ m . c . m = 3 b^{2} (a - b) \end{matrix}$
$\begin{matrix} x^{2} y, x^{2} y + x y^{2} \\ x^{2} y \\ x^{2} y + x y^{2} = x y (x + y) \\ m . c . m = x^{2} y (x + y) \end{matrix}$
$\begin{matrix} 8, 4 + 8 a \\ 8 = 2^{3} \\ 4 + 8 a = 2^{2} (1 + 2 a) \\ m . c . m = 2^{3} (1 + 2 a) \\ m . c . m = 8 (1 + 2 a) \end{matrix}$
$\begin{matrix} 6 a^{2} b, 3 a^{2} b^{2} + 6 a b^{3} \\ 6 a^{2} b = 2.3 a^{2} b \\ 3 a^{2} b^{2} + 6 a b^{3} = 3 a b^{2} (a + 2 b) \\ m . c . m = 2.3 a^{2} b^{2} (a + 2 b) \\ m . c . m = 6 a^{2} b^{2} (a + 2 b) \end{matrix}$
$\begin{matrix} 14 x^{2}, 6 x^{2} + 4 x y \\ 14 x^{2} = 2.7 x^{2} \\ 6 x^{2} + 4 x y = 2 x (3 x + 2 y) \\ m . c . m = 2.7 x^{2} (3 x + 2 y) \\ m . c . m = 14 x^{2} (3 x + 2 y) \end{matrix}$
$\begin{matrix} 9 m, 6 m n^{2} - 12 m n \\ 9 m = 3^{2} m \\ 6 m n^{2} - 12 m n = 6 m n (n - 2) = 2.3 m n (n - 2) \\ m . c . m = 2. 3^{2} m n (n - 2) \\ m . c . m = 18 m n (n - 2) \end{matrix}$
$\begin{matrix} 15, 3 x + 6 \\ 15 = 3.5 \\ 3 x + 6 = 3 (x + 2) \\ m . c . m = 3.5 (x + 2) \\ m . c . m = 15 (x + 2) \end{matrix}$
$\begin{matrix} 10, 5 - 15 b \\ 10 = 2.5 \\ 5 - 15 b = 5 (1 - 3 b) \\ m . c . m = 2.5 (1 - 3 b) \\ m . c . m = 10 (1 - 3 b) \end{matrix}$
$\begin{matrix} 36 a^{2}, 4 a x - 12 a y \\ 36 a^{2} = 2^{2} . 3^{2} a^{2} \\ 4 a x - 12 a y = 4 a (x - 3 y) = 2^{2} a (x - 3 y) \\ m . c . m = 2^{2} . 3^{2} a^{2} (x - 3 y) \\ m . c . m = 36 a^{2} (x - 3 y) \end{matrix}$
$\begin{matrix} 12 x y^{2}, 2 a x^{2} y^{3} + 5 x^{2} y^{3} \\ 12 x y^{2} = 2^{2} .3 x y^{2} \\ 2 a x^{2} y^{3} + 5 x^{2} y^{3} = x^{2} y^{3} (2 a + 5) \\ m . c . m = 2^{2} .3 x^{2} y^{3} (2 a + 5) \\ m . c . m = 12 x^{2} y^{3} (2 a + 5) \end{matrix}$
$\begin{matrix} m n, m^{2}, m n^{3} - m n^{2} \\ m n \\ m^{2} \\ m n^{3} - m n^{2} = m n^{2} (n - 1) \\ m . c . m = m^{2} n^{2} (n - 1) \end{matrix}$
$\begin{matrix} 2 a^{2}, 6 a b, 3 a^{2} - 6 a b \\ 2 a^{2} \\ 6 a b = 2.3 a b \\ 3 a^{2} - 6 a b = 3 a (a - 2 b) \\ m . c . m = 2.3 a^{2} b (a - 2 b) \\ m . c . m = 6 a^{2} b (a - 2 b) \end{matrix}$
$\begin{matrix} x y^{2}, x^{2} y^{3}, 5 x^{5} - 5 x^{4} \\ x y^{2} \\ x^{2} y^{3} \\ 5 x^{5} - 5 x^{4} = 5 x^{4} (x - 1) \\ m . c . m = 5 x^{4} y^{3} (x - 1) \end{matrix}$
$\begin{matrix} 9 a^{2}, 18 b^{3}, 27 a^{4} b + 81 a^{3} b^{2} \\ 9 a^{2} = 3^{2} a^{2} \\ 18 b^{3} = 2. 3^{2} b^{3} \\ 27 a^{4} b + 81 a^{3} b^{2} = 27 a^{3} b (a + 3 b) = 3^{3} a^{3} b (a + 3 b) \\ m . c . m = 2. 3^{3} a^{3} b^{3} (a + 3 b) \\ m . c . m = 54 a^{3} b^{3} (a + 3 b) \end{matrix}$
$\begin{matrix} 10, 6 x^{2}, 9 x^{3} y + 9 x y^{3} \\ 10 = 2.5 \\ 6 x^{2} = 2.3 x^{2} \\ 9 x^{3} y + 9 x y^{3} = 9 x y (x^{2} + y^{2}) = 3^{2} x y (x^{2} + y^{2}) \\ m . c . m = 2. 3^{2} .5 x y (x^{2} + y^{2}) \\ m . c . m = 90 x y (x^{2} + y^{2}) \end{matrix}$
$\begin{matrix} 4 x, x^{3} + x^{2}, x^{2} y - x y \\ 4 x = 2^{2} x \\ x^{3} + x^{2} = x^{2} (x + 1) \\ x^{2} y - x y = x y (x - 1) \\ m . c . m = 2^{2} x^{2} y (x - 1) (x + 1) \\ m . c . m = 4 x^{2} y (x - 1) (x + 1) \end{matrix}$
$\begin{matrix} 24, 6 m^{2} + 18 m, 8 m - 24 \\ 24 = 2^{3} .3 \\ 6 m^{2} + 18 m = 6 m (m + 3) = 2.3 m (m + 3) \\ 8 m - 24 = 8 (m - 3) = 2^{3} (m - 3) \\ m . c . m = 2^{3} .3 m (m + 3) (m - 3) \\ m . c . m = 24 m (m + 3) (m - 3) \end{matrix}$
$\begin{matrix} 2 a^{2} b^{2}, 3 a x + 3 a, 6 x - 18 \\ 2 a^{2} b^{2} \\ 3 a x + 3 a = 3 a (x + 1) \\ 6 x - 18 = 6 (x - 3) = 2.3 (x - 3) \\ m . c . m = 2.3 a^{2} b^{2} (x + 1) (x - 3) \\ m . c . m = 6 a^{2} b^{2} (x + 1) (x - 3) \end{matrix}$
$\begin{matrix} x^{2}, x^{3} + x^{2} - 2 x, x^{2} + 4 x + 4 \\ x^{2} \\ x^{3} + x^{2} - 2 x = x (x^{2} + x - 2) = x (x + 2) (x - 1) \\ x^{2} + 4 x + 4 = {(x + 2)}^{2} \\ m . c . m = x^{2} {(x + 2)}^{2} (x - 1) \end{matrix}$
$\begin{matrix} 6 a b, x^{2} - 4 x y + 4 y^{2}, 9 a^{2} x - 18 a^{2} y \\ 6 a b = 2.3 a b \\ x^{2} - 4 x y + 4 y^{2} = {(x - 2 y)}^{2} \\ 9 a^{2} x - 18 a^{2} y = 9 a^{2} (x - 2 y) = 3^{2} a^{2} (x - 2 y) \\ m . c . m = 2. 3^{2} a^{2} {(x - 2 y)}^{2} \\ m . c . m = 18 a^{2} {(x - 2 y)}^{2} \end{matrix}$
$\begin{matrix} 6 x^{3}, 3 x^{3} - 3 x^{2} - 18 x, 9 x^{4} - 36 x^{2} \\ 6 x^{3} = 2.3 x^{3} \\ 3 x^{3} - 3 x^{2} - 18 x = 3 x (x^{2} - x - 6) = 3 x (x - 3) (x + 2) \\ 9 x^{4} - 36 x^{2} = 9 x^{2} (x^{2} - 4) = 3^{2} x^{2} (x + 2) (x - 2) \\ m . c . m = 2. 3^{2} x^{2} (x - 3) (x + 2) (x - 2) \\ m . c . m = 18 x^{2} (x - 3) (x + 2) (x - 2) \end{matrix}$
$\begin{matrix} a^{2} x^{2}, 4 x^{3} - 12 x^{2} y + 9 x y^{2}, 2 x^{4} - 3 x^{3} y \\ a^{2} x^{2} \\ 4 x^{3} - 12 x^{2} y + 9 x y^{2} = x (4 x^{2} - 12 x y + 9 y^{2}) = x {(2 x - 3 y)}^{2} \\ 2 x^{4} - 3 x^{3} y = x^{3} (2 x - 3 y) \\ m . c . m = a^{2} x^{3} {(2 x - 3 y)}^{2} \end{matrix}$
$\begin{matrix} 8 x^{3}, 12 x^{2} y^{2}, 9 x^{2} - 45 x \\ 8 x^{3} = 2^{3} x^{3} \\ 12 x^{2} y^{2} = 2^{2} .3 x^{2} y^{2} \\ 9 x^{2} - 45 x = 9 x (x - 5) = 3^{2} x (x - 5) \\ m . c . m = 2^{3} . 3^{2} x^{3} y^{2} (x - 5) \\ m . c . m = 72 x^{3} y^{2} (x - 5) \end{matrix}$
$\begin{matrix} a n^{3}, 2 n, n^{2} x^{2} + n^{2} y^{2}, n x^{2} + 2 n x y + n y^{2} \\ a n^{3} \\ 2 n \\ n^{2} x^{2} + n^{2} y^{2} = n^{2} (x^{2} + y^{2}) \\ n x^{2} + 2 n x y + n y^{2} = n (x^{2} + 2 x y + y^{2}) = n {(x + y)}^{2} \\ m . c . m = 2 a n^{3} (x^{2} + y^{2}) {(x + y)}^{2} \end{matrix}$
$\begin{matrix} 8 x^{2}, x^{3} + x^{2} - 6 x, 2 x^{3} - 8 x^{2} + 8 x, 4 x^{3} + 24 x^{2} + 36 x \\ 8 x^{2} = 2^{3} x^{2} \\ x^{3} + x^{2} - 6 x = x (x^{2} + x - 6) = x (x + 3) (x - 2) \\ 2 x^{3} - 8 x^{2} + 8 x = 2 x (x^{2} - 4 x + 4) = 2 x {(x - 2)}^{2} \\ 4 x^{3} + 24 x^{2} + 36 x = 4 x (x^{2} + 6 x + 9) = 2^{2} x {(x + 3)}^{2} \\ m . c . m = 2^{3} x^{2} {(x - 2)}^{2} {(x + 3)}^{2} \\ m . c . m = 8 x^{2} {(x - 2)}^{2} {(x + 3)}^{2} \end{matrix}$
$\begin{matrix} 3 x^{3}, x^{3} + 1, 2 x^{2} - 2 x + 2, 6 x^{3} + 6 x^{2} \\ 3 x^{3} \\ x^{3} + 1 = (x + 1) (x^{2} - x + 1) \\ 2 x^{2} - 2 x + 2 = 2 (x^{2} - x + 1) \\ 6 x^{3} + 6 x^{2} = 6 x^{2} (x + 1) = 2.3 x^{2} (x + 1) \\ m . c . m = 2.3 x^{2} (x + 1) (x^{2} - x + 1) \\ m . c . m = 6 x^{2} (x + 1) (x^{2} - x + 1) \end{matrix}$
$\begin{matrix} 4 x y^{2}, 3 x^{3} - 3 x^{2}, a^{2} + 2 a b + b^{2}, a x - a + b x - b \\ 4 x y^{2} = 2^{2} x y^{2} \\ 3 x^{3} - 3 x^{2} = 3 x^{2} (x - 1) \\ a^{2} + 2 a b + b^{2} = {(a + b)}^{2} \\ a x - a + b x - b = a (x - 1) + b (x - 1) = (a + b) (x - 1) \\ m . c . m = 2^{2} .3 x^{2} y^{2} {(a + b)}^{2} (x - 1) \\ m . c . m = 12 x^{2} y^{2} {(a + b)}^{2} (x - 1) \end{matrix}$
$\begin{matrix} 2 a, 4 b, 6 a^{2} b, 12 a^{2} - 24 a b + 12 b^{2}, 5 a b^{3} - 5 b^{4} \\ 2 a \\ 4 b = 2^{2} b \\ 6 a^{2} b = 2.3 a^{2} b \\ 12 a^{2} - 24 a b + 12 b^{2} = 12 (a^{2} - 2 a b + b^{2}) = 2^{2} .3 {(a - b)}^{2} \\ 5 a b^{3} - 5 b^{4} = 5 b^{3} (a - b) \\ m . c . m = 2^{2} .3.5 a^{2} b^{3} {(a - b)}^{2} \\ m . c . m = 60 a^{2} b^{3} {(a - b)}^{2} \end{matrix}$
$\begin{matrix} 28 x, x^{2} + 2 x + 1, x^{2} + 1, 7 x^{2} + 7, 14 x + 14 \\ 28 x = 2^{2} .7 x \\ x^{2} + 2 x + 1 = {(x + 1)}^{2} \\ x^{2} + 1 \\ 7 x^{2} + 7 = 7 (x^{2} + 1) \\ 14 x + 14 = 14 (x + 1) \\ m . c . m = 2^{2} .7 x {(x + 1)}^{2} (x^{2} + 1) \\ m . c . m = 28 x {(x + 1)}^{2} (x^{2} + 1) \end{matrix}$