# Ejercicio 115

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CAPITULO XII

Mínimo Común Múltiplo
Ejercicio 115
Hallar el m.c.m de:
1. $\begin{array}{cc}{a}^{2},\phantom{\rule{6px}{0ex}}a{b}^{2}& \\ & m.c.m={a}^{2}{b}^{2}\end{array}$
2. $\begin{array}{cc}{x}^{2}y,\phantom{\rule{6px}{0ex}}x{y}^{2}& \\ & m.c.m={x}^{2}{y}^{2}\end{array}$
3. $\begin{array}{cc}a{b}^{2}c,\phantom{\rule{6px}{0ex}}{a}^{2}bc& \\ & m.c.m={a}^{2}{b}^{2}c\end{array}$
4. $\begin{array}{cc}{a}^{2}{x}^{3},\phantom{\rule{6px}{0ex}}{a}^{3}b{x}^{2}& \\ & m.c.m={a}^{3}b{x}^{3}\end{array}$
5. $\begin{array}{cc}6{m}^{2}n,\phantom{\rule{6px}{0ex}}4{m}^{3}& \\ & \\ & 6{m}^{2}n=2.3{m}^{2}n\\ & 4{m}^{3}={2}^{2}{m}^{3}\\ & \\ & m.c.m={2}^{2}.3{m}^{3}n\\ & m.c.m=12{m}^{3}n\end{array}$
6. $\begin{array}{cc}9a{x}^{3}{y}^{4},\phantom{\rule{6px}{0ex}}15{x}^{2}{y}^{5}& \\ & \\ & 9a{x}^{3}{y}^{4}={3}^{2}a{x}^{3}{y}^{4}\\ & 15{x}^{2}{y}^{5}=3.5{x}^{2}{y}^{5}\\ & \\ & m.c.m={3}^{2}.5a{x}^{3}{y}^{5}\\ & m.c.m=45a{x}^{3}{y}^{5}\end{array}$
7. $\begin{array}{cc}{a}^{3},\phantom{\rule{6px}{0ex}}a{b}^{2},\phantom{\rule{6px}{0ex}}{a}^{2}b& \\ & \\ & m.c.m={a}^{3}{b}^{2}\end{array}$
8. $\begin{array}{cc}{x}^{2}y,\phantom{\rule{6px}{0ex}}x{y}^{2},\phantom{\rule{6px}{0ex}}x{y}^{3}z& \\ & \\ & m.c.m={x}^{2}{y}^{3}z\end{array}$
9. $\begin{array}{cc}2a{b}^{2},\phantom{\rule{6px}{0ex}}4{a}^{2}b,\phantom{\rule{6px}{0ex}}8{a}^{3}& \\ & \\ & 2a{b}^{2}\\ & 4{a}^{2}b={2}^{2}{a}^{2}b\\ & 8{a}^{3}={2}^{3}{a}^{3}\\ & \\ & m.c.m={2}^{3}{a}^{3}{b}^{2}\\ & m.c.m=8{a}^{3}{b}^{2}\end{array}$
10. $\begin{array}{cc}3{x}^{2}{y}^{3}z,\phantom{\rule{6px}{0ex}}4{x}^{3}{y}^{3}{z}^{2},\phantom{\rule{6px}{0ex}}6{x}^{4}& \\ & \\ & 3{x}^{2}{y}^{3}z\\ & 4{x}^{3}{y}^{3}{z}^{2}={2}^{2}{x}^{3}{y}^{3}{z}^{2}\\ & 6{x}^{4}=2.3{x}^{4}\\ & \\ & m.c.m={2}^{2}.3{x}^{4}{y}^{3}{z}^{2}\\ & m.c.m=12{x}^{4}{y}^{3}{z}^{2}\end{array}$
11. $\begin{array}{cc}6m{n}^{2},\phantom{\rule{6px}{0ex}}9{m}^{2}{n}^{3},\phantom{\rule{6px}{0ex}}12{m}^{3}n& \\ & \\ & 6m{n}^{2}=2.3m{n}^{2}\\ & 9{m}^{2}{n}^{3}={3}^{2}{m}^{2}{n}^{3}\\ & 12{m}^{3}n={2}^{2}.3{m}^{3}n\\ & \\ & m.c.m={2}^{2}.{3}^{2}{m}^{3}{n}^{3}\\ & m.c.m=36{m}^{3}{n}^{3}\end{array}$
12. $\begin{array}{cc}3{a}^{2},\phantom{\rule{6px}{0ex}}4{b}^{2},\phantom{\rule{6px}{0ex}}8{x}^{2}& \\ & \\ & 3{a}^{2}\\ & 4{b}^{2}={2}^{2}{b}^{2}\\ & 8{x}^{2}={2}^{3}{x}^{2}\\ & \\ & m.c.m={2}^{3}.3{a}^{2}{b}^{2}{x}^{2}\\ & m.c.m=24{a}^{2}{b}^{2}{x}^{2}\end{array}$
13. $\begin{array}{cc}5{x}^{2},\phantom{\rule{6px}{0ex}}10xy,\phantom{\rule{6px}{0ex}}15x{y}^{2}& \\ & \\ & 5{x}^{2}\\ & 10xy=2.5xy\\ & 15x{y}^{2}=3.5x{y}^{2}\\ & \\ & m.c.m=2.3.5{x}^{2}{y}^{2}\\ & m.c.m=30{x}^{2}{y}^{2}\end{array}$
14. $\begin{array}{cc}a{x}^{3}{y}^{2},\phantom{\rule{6px}{0ex}}{a}^{3}xy,\phantom{\rule{6px}{0ex}}{a}^{2}{x}^{2}{y}^{3}& \\ & \\ & m.c.m={a}^{3}{x}^{3}{y}^{3}\end{array}$
15. $\begin{array}{cc}4ab,\phantom{\rule{6px}{0ex}}6{a}^{2},\phantom{\rule{6px}{0ex}}3{b}^{2}& \\ & \\ & 4ab={2}^{2}ab\\ & 6{a}^{2}=2.3{a}^{2}\\ & 3{b}^{2}\\ & \\ & m.c.m={2}^{2}.3{a}^{2}{b}^{2}\\ & m.c.m=12{a}^{2}{b}^{2}\end{array}$
16. $\begin{array}{cc}3{x}^{3},\phantom{\rule{6px}{0ex}}6{x}^{2},\phantom{\rule{6px}{0ex}}9{x}^{4}{y}^{2}& \\ & \\ & 3{x}^{3}\\ & 6{x}^{2}=2.3{x}^{2}\\ & 9{x}^{4}{y}^{2}={3}^{2}{x}^{4}{y}^{2}\\ & \\ & m.c.m=2.{3}^{2}{x}^{4}{y}^{2}\\ & m.c.m=18{x}^{4}{y}^{2}\end{array}$
17. $\begin{array}{cc}9{a}^{2}bx,\phantom{\rule{6px}{0ex}}12a{b}^{2}{x}^{2},\phantom{\rule{6px}{0ex}}18{a}^{3}{b}^{3}x& \\ & \\ & 9{a}^{2}bx={3}^{2}{a}^{2}bx\\ & 12a{b}^{2}{x}^{2}=3.4a{b}^{2}{x}^{2}\\ & 18{a}^{3}{b}^{3}x=2.{3}^{2}{a}^{3}{b}^{3}x\\ & \\ & m.c.m=2.{3}^{2}.4{a}^{3}{b}^{3}{x}^{2}\\ & m.c.m=72{a}^{3}{b}^{3}{x}^{2}\end{array}$
18. $\begin{array}{cc}10{m}^{2},\phantom{\rule{6px}{0ex}}15m{n}^{2},\phantom{\rule{6px}{0ex}}20{n}^{3}& \\ & \\ & 10{m}^{2}=2.5{m}^{2}\\ & 15m{n}^{2}=3.5m{n}^{2}\\ & 20{n}^{3}={2}^{2}.5{n}^{3}\\ & \\ & m.c.m={2}^{2}.3.5{m}^{2}{n}^{3}\\ & m.c.m=60{m}^{2}{n}^{3}\end{array}$
19. $\begin{array}{cc}18{a}^{3},\phantom{\rule{6px}{0ex}}24{b}^{2},\phantom{\rule{6px}{0ex}}36a{b}^{3}& \\ & \\ & 18{a}^{3}=2.{3}^{2}{a}^{3}\\ & 24{b}^{2}={2}^{3}.3{b}^{2}\\ & 36a{b}^{3}={2}^{2}.{3}^{2}a{b}^{3}\\ & \\ & m.c.m={2}^{3}.{3}^{2}{a}^{3}{b}^{3}\\ & m.c.m=72{a}^{3}{b}^{3}\end{array}$
20. $\begin{array}{cc}20{m}^{2}{n}^{3},\phantom{\rule{6px}{0ex}}24{m}^{3}n,\phantom{\rule{6px}{0ex}}30m{n}^{2}& \\ & \\ & 20{m}^{2}{n}^{3}={2}^{2}.5{m}^{2}{n}^{3}\\ & 24{m}^{3}n={2}^{3}.3{m}^{3}n\\ & 30m{n}^{2}=2.3.5m{n}^{2}\\ & \\ & m.c.m={2}^{3}.3.5{m}^{3}{n}^{3}\\ & m.c.m=120{m}^{3}{n}^{3}\end{array}$
21. $\begin{array}{cc}a{b}^{2},\phantom{\rule{6px}{0ex}}b{c}^{2},\phantom{\rule{6px}{0ex}}{a}^{2}{c}^{3},\phantom{\rule{6px}{0ex}}{b}^{3}{c}^{3}& \\ & \\ & m.c.m={a}^{2}{b}^{3}{c}^{3}\end{array}$
22. $\begin{array}{cc}2{x}^{2}y,\phantom{\rule{6px}{0ex}}8x{y}^{3},\phantom{\rule{6px}{0ex}}4{a}^{2}{x}^{3},\phantom{\rule{6px}{0ex}}12{a}^{3}& \\ & \\ & 2{x}^{2}y\\ & 8x{y}^{3}={2}^{3}x{y}^{3}\\ & 4{a}^{2}{x}^{3}={2}^{2}{a}^{2}{x}^{3}\\ & 12{a}^{3}={2}^{2}.3{a}^{3}\\ & \\ & m.c.m={2}^{3}.3{a}^{3}{x}^{3}{y}^{3}\\ & m.c.m=24{a}^{3}{x}^{3}{y}^{3}\end{array}$
23. $\begin{array}{cc}6{a}^{2},\phantom{\rule{6px}{0ex}}9x,\phantom{\rule{6px}{0ex}}12a{y}^{2},\phantom{\rule{6px}{0ex}}18{x}^{3}y& \\ & \\ & 6{a}^{2}=2.3{a}^{2}\\ & 9x={3}^{2}x\\ & 12a{y}^{2}={2}^{2}.3a{y}^{2}\\ & 18{x}^{3}y=2.{3}^{2}{x}^{3}y\\ & \\ & m.c.m={2}^{2}.{3}^{2}{a}^{2}{x}^{3}{y}^{2}\\ & m.c.m=36{a}^{2}{x}^{3}{y}^{2}\end{array}$
24. $\begin{array}{cc}15m{n}^{2},\phantom{\rule{6px}{0ex}}10{m}^{2},\phantom{\rule{6px}{0ex}}20{n}^{3},\phantom{\rule{6px}{0ex}}25m{n}^{4}& \\ & \\ & 15m{n}^{2}=3.5m{n}^{2}\\ & 10{m}^{2}=2.5{m}^{2}\\ & 20{n}^{3}={2}^{2}.5{n}^{3}\\ & 25m{n}^{4}={5}^{2}m{n}^{4}\\ & \\ & m.c.m={2}^{2}.3.{5}^{2}{m}^{2}{n}^{4}\\ & m.c.m=300{m}^{2}{n}^{4}\end{array}$
25. $\begin{array}{cc}24{a}^{2}{x}^{3},\phantom{\rule{6px}{0ex}}36{a}^{2}{y}^{4},\phantom{\rule{6px}{0ex}}40{x}^{2}{y}^{5},\phantom{\rule{6px}{0ex}}60{a}^{3}{y}^{6}& \\ & \\ & 24{a}^{2}{x}^{3}={2}^{3}.3{a}^{2}{x}^{3}\\ & 36{a}^{2}{y}^{4}={2}^{2}.{3}^{2}{a}^{2}{y}^{4}\\ & 40{x}^{2}{y}^{5}={2}^{3}.5{x}^{2}{y}^{5}\\ & 60{a}^{3}{y}^{6}={2}^{2}.3.5{a}^{3}{y}^{6}\\ & \\ & m.c.m={2}^{3}.{3}^{2}.5{a}^{3}{x}^{3}{y}^{6}\\ & m.c.m=360{a}^{3}{x}^{3}{y}^{6}\end{array}$
26. $\begin{array}{cc}3{a}^{3},\phantom{\rule{6px}{0ex}}8ab,\phantom{\rule{6px}{0ex}}10{b}^{2},\phantom{\rule{6px}{0ex}}12{a}^{2}{b}^{3},\phantom{\rule{6px}{0ex}}16{a}^{2}{b}^{2}& \\ & \\ & 3{a}^{3}\\ & 8ab={2}^{3}ab\\ & 10{b}^{2}=2.5{b}^{2}\\ & 12{a}^{2}{b}^{3}={2}^{2}.3{a}^{2}{b}^{3}\\ & 16{a}^{2}{b}^{2}={2}^{4}{a}^{2}{b}^{2}\\ & \\ & m.c.m={2}^{4}.3.5{a}^{3}{b}^{3}\\ & m.c.m=240{a}^{3}{b}^{3}\end{array}$