Question

.Find the remainder when 3^{53} is divided by 10?

Answer 1

Answer:

1.93832457e24 is it ? .or not

Answer 2

Remainder Theorem

Given:

dividend= $3^{53}$

divisor= 10

To find:

remainder when given dividend is divided by divisor

Explanation:

Remainder Theorem:

It states that when a polynomial is p(x) is divided by binomial x-a , the remainder obtained is p(a).

The formula :

Dividend= (divisor * quotient) + remainder

Steps:

1. First leave the power aside and divide the base value of both dividend and divisor.
2. So, 3 divided by 10 will give remainder as 3 or -7.
3. Calculations:      

                  $\frac{3^{1} }{10} =3 or -7\\\\\frac{3^{2} }{10} =9 or -1\\\\\frac{3^{3} }{10} =7\\\\\frac{3^{4} }{10} =1\\\\\frac{3^{5} }{10} =3$

   4. We can write in this form also,

               $\frac{3^{53} }{10}=\frac{({3^{2}})^{25}*3^3}{10}$

   5.  As, $\frac{3^{2} }{10}=-1$,                {∵$(a^{m})^n = a^{m*n}$    and     $a^{m}*a^{n} = a^{m+n}$}

         $\frac{3^{3}*(-1)^{25}}{10} \\=-27/10\\So, -7 or 3 .$

Hence the remainder  will be 3.

This can also be evaluated using Long division or Euclid division algorithm (to find GCD of 2 numbers divide one of them into the another, now again divide old remainder into the old divisor, continue this till remainder is 0, the last non zero remainder will be the GCD of two numbers)