Question

Find Reminder ??
when
${2}^{256} devided \: by \: 17$

Answer 1

Hey , there !!!

this is Fermat's theorem of power .

Remainders follow a cyclic pattern. For example: powers of 4 always have 1 as it's remainder when divided by 3. 4, 16, 64, 256, 1024..

Here the powers of 2 and their remainders when divided by 17 is listed.

Powers (Remainders)

1 (2)

2 (4)

3 (8)

4 (16)

5 (32–17=15)

6 (15*2–17=13)

7 (13*2–17=9)

8 (9*2–17=1)

9 (1*2=2)

.

.

.

So the cyclic pattern here is: 2, 4, 8, 16, 15, 13, 9, 1 and repeat. So, 2^8= 256 when divided by 17 leaves out a remainder of 1.

Now, 2^(256)= 2^(8*32)

=(2^8)^32

=256^32

Since remainder of 256 is 1. And powers of 256/17 is always one. So, the answer is 1.

hope it definitely helps u Amrit !!

be happy and keep smiling !!!
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# himanshu Jha ✌✌