Q. 184.5( 4 Votes )

Show that p – 1 is a factor of p10 – 1 and also of p11 – 1.

Answer :

let h (p) = 𝑝 10 − 1,and g(p) = 𝑝 − 1


Putting g (p) = 0 𝑝 − 1 = 0 𝑝 = 1


According to factor theorem if g(p) is a factor of h(p) , then h(1) should be zero


h(1) = (1)10 − 1 = 1 − 1 = 0


g (p) is a factor of h(p).


Now, we have h (p) = 𝑝 11 − 1, g (p) = 𝑝 − 1


Putting g (p) = 0 𝑝 − 1 = 0 𝑝 = 1


According to factor theorem if g (p) is a factor of h (p) ,


Then h(1) = 0


(1)11 – 1 = 0 hence g (p) the factor of h (p)


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