Question

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Answer 1

Solution

(p-1) is given we need to establish that its a factor of $p^{10}\:-\:1$ & $p^{11}\:-\:1$

Let, f(p) = $p^{10}\:-\:1$

f(1) = $(1)^{10}\:-\:1$

f(1) = $(1)\:-\:1$

f(1) = $0$

That means (p-1) is a factor of equation f(p)

Let, r(p) = $p^{11}\:-\:1$

r(1) = $(1)^{11}\:-\:1$

r(1) = $(1)\:-\:1$

r(1) = $0$

That means (p-1) is also a factor of equation r(p)

We checked and found (p-1) is a factor of both equations.

Hence Proved

Answer 2

Answer:

Solution

(p-1) is given we need to establish that its a factor of p^{10}\:-\:1p

10

−1 & p^{11}\:-\:1p

11

−1

Let, f(p) = p^{10}\:-\:1p

10

−1

f(1) = (1)^{10}\:-\:1(1)

10

−1

f(1) = (1)\:-\:1(1)−1

f(1) = 00

That means (p-1) is a factor of equation f(p)

Let, r(p) = p^{11}\:-\:1p

11

−1

r(1) = (1)^{11}\:-\:1(1)

11

−1

r(1) = (1)\:-\:1(1)−1

r(1) = 00

That means (p-1) is also a factor of equation r(p)

We checked and found (p-1) is a factor of both equations.

Hence Proved