Question

the remainder when 3^1999 is divided 47​

Answer 1

Answer: 21

Given: $3^{1999}$ is the number given.

To Find: we have to find the remainder when the given number is divided by 47

Step-by-Step Explanation:

Step 1:

Firstly, we will see the definition of mod.

Suppose, there are two numbers a and b then a mod b is equal to the remainder that is obtained when a is divided by b.

We will see two properties of remainders that will help us solve this question:

Property 1: (x*y) mod p = ((x mod p)*(y mod p)) mod p

For Example: (12*8) mod 5 = 96 mod 5 = 1

But when we use property 1, it is ((12mod5)*(8mod5)) mod 5

                                                      = (2*3)mod5

                                                       = 6 mod 5 = 1

Property 2: Fermat's Theorem

$a^{p-1}$ mod p = 1

where p is a prime and hcf(a,p) = 1

Step 2:

$3^{1999}$ mod 47 = ($3^{46}$* $3^{46}$ * ........*$3^{46}$(43 times) * $3^{21}$ ) mod 47

=( $3^{46}$ mod 47 * $3^{46}$ mod 47.... $3^{46}$mod 47(43 times) * $3^{21}$ mod 47) mod47

= (1*1*1.......*1(43 times)* $3^{21}$) mod 47 ( by property 2)

= $3^{21}$mod 47

= ($(3^{5})^4$* 3) mod 47

= ($243^{4}$ * 3) mod 47

= (243*243*243*243*3) mod 47

= (243 mod 47* 243 mod 47*243 mod 47*243 mod 47 * 3 mod 47) mod 47

= (8 mod 47 * 8 mod 47 * 8 mod 47 * 8 mod 47 * 3 mod 47) mod 47

= (8*8*8*8*3) mod 47

= ( 64 * 64 * 3) mod 47

= (64*192) mod 47

= (64 mod 47 * 192 mod 47

= (17 *4) mod 47

= 68 mod 47

= 21

To know more about remainders, refer to the links below:

https://brainly.in/question/38482403

https://brainly.in/question/41952097

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