Respuesta :
Answer:
Proved
Step-by-step explanation:
Given
Integers: a, b, c
Required
Show that c is a multiple of a⁶
First, we need to list out the multiples of a³
[tex]a^3 -> a^3, a^6, a^9, a^{12}...[/tex]
Since b is a multiple of a³, then b can be any of the listed multiples
Next, is to list out the multiples of b²
[tex]b^2 -> b^2,b^4,b^6,b^8,b^{10}....[/tex]
Since c is a multiple of b², then c can be any of the listed multiples
To list out the multiples of a⁶, we have to get the common multiples of a³ and b² in terms of a
Substitute a³ for b in
[tex]b^2 -> b^2,b^4,b^6,b^8,b^{10}....[/tex]
[tex](a^3)^2 -> (a^3)^2,(a^3)^4,(a^3)^6,(a^3)^8,(a^3)^{10}....[/tex]
[tex]a^6 -> a^6,a^{12},a^{18},a^{24},a^{30}....[/tex]
Recall that c is a multiple of b²
From the above listed multiples, we have a⁶ listed as one of the multiple; Hence, c is a multiple of a⁶
Yes C should be multiple of [tex]a^6[/tex]
Calculation of the multiple:
Since it is mentioned that b should be the multiple of [tex]a^3[/tex]
So,
[tex]b = a^3R[/tex] here some integer should be R
And, the c should be multiple of [tex]b^2[/tex]
So,
[tex]C = b^2 r[/tex] for some integer r
Now we have to substitute the value of b from the first equation
i.e.
[tex]C = (a^3R)^2r = a^6 (R^2r)[/tex]
Here [tex]R^2r[/tex] is some integer
so,
[tex]C = a^6R'[/tex]
Therefore, we can say that Yes C should be multiple of [tex]a^6[/tex]
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