arnavpx arnavpx
  • 02-02-2020
  • Mathematics
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Prove the divisibility of the following numbers:
24^54·54^24·2^10 by 72^63

Respuesta :

LammettHash
LammettHash LammettHash
  • 02-02-2020

Factorize each composite base:

[tex]24=2^3\cdot3[/tex]

[tex]54=2\cdot3^3[/tex]

[tex]72=2^3\cdot3^2[/tex]

Then

[tex]24^{54}=2^{162}\cdot3^{54}[/tex]

[tex]54^{24}=2^{24}\cdot3^{72}[/tex]

[tex]72^{63}=2^{189}\cdot3^{126}[/tex]

Now cancel as many factors as possible:

[tex]\dfrac{24^{54}\cdot54^{24}\cdot2^{10}}{72^{63}}=\dfrac{2^{196}\cdot3^{126}}{2^{189}\cdot3^{126}}=2^7[/tex]

The denominator vanishes, so [tex]72^{63}[/tex] does divide [tex]24^{54}\cdot54^{24}\cdot2^{10}[/tex].

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