Question

if x^51 + 51 is divided by x+1 then remainder is

Answer 1

The remainder is 50.

One way to see it is through smartly expanding x51+51x51+51. The approach is as follows:

x51+51=x51+(x50−x50)+(−x49+x49)+(x48−x48)+...+(x2−x2)+(−x+x)+1+50x51+51=x51+(x50−x50)+(−x49+x49)+(x48−x48)+...+(x2−x2)+(−x+x)+1+50

Now, regrouping the terms gives,
x51+51=(x51+x50)−(x50+x49)+(x49+x48)−x48+...+x2−(x2+x)+(x+1)+50x51+51=(x51+x50)−(x50+x49)+(x49+x48)−x48+...+x2−(x2+x)+(x+1)+50

=x50(x+1)−x49(x+1)+x48(x+1)−...−x(x+1)+(x+1)+50=x50(x+1)−x49(x+1)+x48(x+1)−...−x(x+1)+(x+1)+50

=(x+1)(x50−x49+x48...−x+1)+50=(x+1)(x50−x49+x48...−x+1)+50

So, x51+51=(x+1)p(x)+50x51+51=(x+1)p(x)+50, where p(x) = (x50−x49+x48...−x+1)(x50−x49+x48...−x+1)

Now, does the above expression ring a bell about what we learnt in school which is something like:

Dividend=(Divisor∗Quotient)+RemainderDividend=(Divisor∗Quotient)+Remainder

Mapping this representation to the above case results in as follows:
Dividend=x51+51Dividend=x51+51

Divisor=x+1Divisor=x+1

Quotient=p(x)=(x50−x49+x48...−x+1)Quotient=p(x)=(x50−x49+x48...−x+1)

Remainder=50Remainder=50

Another way to see this is through the remainder theorem,

Let's say f(x)=x51+51f(x)=x51+51, then the remainder rr, obtained by dividing f(x)f(x) with x+1=x−(−1)x+1=x−(−1) is f(−1)f(−1).

Hence, r=f(−1)=(−1)51+51=−1+51=50