Question

Challenge for You
A natural number n(n\geq 2)\text{,}n(n≥2), always satisfy the equation
\boxed{x^{n}-1=(x-1)(x^{n-1}+x^{n-2}+\cdots+x+1)\text{.}}
Find the remainder when x^{15}+x^{14}+x^{13}+x^{12}+x+1 is divided by x^{4}+x^{3}+x^{2}+x+1\text{.}

Answer 1

Answer:

1 to 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

25 101 - 200 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Answer 2

Given Question :-

$\sf \: A \: natural \: number \: n \geqslant 2 \: satisfy \: the \: equation$

$\red{ \boxed{ \sf{ \: {x}^{n} - 1 = (x - 1)( {x}^{n - 1} + {x}^{n - 2} + - - - + {x}^{3} + {x}^{2} + x + 1)}}}$

Find the remainder when

$\sf \: {x}^{15} + {x}^{14} + {x}^{13} + {x}^{12} + x + 1 \: is \: divided \: by \: {x}^{4} + {x}^{3} + {x}^{2} + x + 1$

Note :- Don't use Long Division.

$\large\underline{\sf{Solution-}}$

Given polynomial is

$\rm :\longmapsto\: {x}^{15} + {x}^{14} + {x}^{13} + {x}^{12} + x + 1$

and other polynomial is

$\rm :\longmapsto\: {x}^{4} + {x}^{3} + {x}^{2} + x + 1$

Now, we have to find the remainder when

$\rm :\longmapsto\:\dfrac{ {x}^{15} + {x}^{14} + {x}^{13} + {x}^{12} + x + 1}{ {x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

can be rewritten as

$\rm \:  =  \: \dfrac{{x}^{15} + {x}^{14} + {x}^{13} + {x}^{12} + {x}^{11} - {x}^{11} + x + 1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

$\rm \:  =  \: \dfrac{({x}^{15} + {x}^{14} + {x}^{13} + {x}^{12} + {x}^{11}) + x - {x}^{11} + 1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

$\rm \:  =  \: \dfrac{ {x}^{11} ({x}^{4} + {x}^{3} + {x}^{2} + {x}^{} + 1) + x(1 - {x}^{10}) + 1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

$\rm \:  =  \:\dfrac{{x}^{11}({x}^{4} + {x}^{3} + {x}^{2} + x + 1)}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1} - \dfrac{ x({x}^{10} - 1) + 1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

$\rm \:  =  {x}^{11} - \dfrac{ x({x}^{5} - 1)( {x}^{5} + 1)}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1} + \dfrac{1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

Now,

$\red{ \boxed{ \sf{ \: {x}^{5} - 1 = (x - 1)({x}^{4} + {x}^{3} + {x}^{2} + x + 1)}}} \\ \\ \red{\boxed{ \sf{ \: \frac{ {x}^{5} - 1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1} = x - 1}}} \: \: \: \: \: \: \: \: \: \: \: \: \:$

So, using this, above can be rewritten as

$\rm \:  =  \: {x}^{11} - x( {x}^{5} + 1)(x - 1) + \dfrac{1}{{x}^{4} + {x}^{3} + {x}^{2} + x + 1}$

So, when

$\rm :\longmapsto\: {x}^{15} + {x}^{14} + {x}^{13} + {x}^{12} + x + 1$

is divided by

$\rm :\longmapsto\:{x}^{4} + {x}^{3} + {x}^{2} + x + 1$

Then,

$\bf\implies \:Remainder \: is \: 1$