Question

find the remainder when 2^81 is divided by 17​

Answer 1

Answer:

Thus reminder is equal to 2 when 2^81 is divided by 17

Step-by-step explanation:

Solution:

We are going to using congruence relation to find the remainder

We know that

    2^5 = 32

⇒  2^5 = 15(mod 17)

Taking square on both sides we get

⇒  (2^5)² = 15²(mod 17)

⇒  2^10 = 225(mod 17)

And 225 = 4(mod 17)    So

⇒    2^10 = 4(mod 17)

Taking square on both sides we get

⇒     (2^10)² = 4²(mod 17)

⇒       2^20 = 16(mod 17)

Taking square on both sides we get

⇒   (2^20)² = (16)²(mod 17)

⇒       2^40 = 256(mod 17)

And     256 = 1(mod 17)  So

⇒       2^40 = 1 (mod 17)

Taking square on both sides we get

⇒   (2^40)² = 1² (mod 17)

⇒      2^80 = 1 (mod 17)

Multiplying by 2 on both sides we get

 (2^80) × 2 = 1 × 2 (mod 17)

⇒        2^81 = 2 (mod 17)

Thus reminder is equal to 2 when 2^81 is divided by 17

Answer 2

Answer:

Remainder when $2^{81}$ divided by 17 is 2.

Step-by-step explanation:

We need to find remainder when $2^{81}$ divided by 17.

So, we can write given number as,

$2^{80}\times2$

$=(2^4)^{20}\times2$

$\imples(16)^{20}\times2$

         

When 16 is divided by 17 it leave -1 as remainder,

$\implies(-1)^{20}\times2$

$\implies1\times2$

⇒ 2

Now, when dividend is smaller than divisor then remainder is equal to dividend.

So, when 2 is divided by 17 it leaves remainder = 2.

Therefore, Remainder when $2^{81}$ divided by 17 is 2.