Question

Find the remainder when 800^45 is divided by 23
Select one:
a. 18
O b.5
O c. 22
O d. 11

Answer 1

Answer:

wait a minute I will give you the ans

Answer 2

Answer:

The correct answer is option a. 18

Given problem

Find the remainder when 800^45 is divided by 23  

Explanation:

given dividend = 800⁴⁵  

         divisor = 23  (which is a prime number)  

first of all simplify 800⁴⁵ for that divide 800 by 23  

when 800 is divided by 23  the reminder will be 18

therefore the reminder when 800⁴⁵ is divided by 23 is same as the reminder when 18⁴⁵ is divisible by 23

now we will find the reminder when 18⁴⁵ is divided by 23  

here 18⁴⁵ can be written as  18²²⁺²²⁺¹  which is equals to 18²²× 18²³ ×18¹

given problem is  =  $\frac{18^{22} . 18^{22} . 18^{1} }{23}$            

from Fermat's little theorem if a and p are co prime numbers and if  $a^{p-1}$ is divisible by p then the reminder will be one

⇒      $\frac{a^{p-1} }{p}$  here reminder will be 1

⇒ when   18²² is divided by 23   reminder = 1    

⇒ and when 18 is divided by 23 reminder =  18

⇒ reminder when  $\frac{18^{22} . 18^{22} . 18^{1} }{23}$   =  1×1×18 = 18

therefore the reminder when 800⁴⁵ is divisible by 23 = 18