Question

2^33 divided by 17 ,then the remainder is.... 1)15. 2)2. 3)4. 4)9. answer with reason

Answer 1

2 leaves remainder 2 on division by 17. This can be written as,

$2\equiv2\pmod{17}$

Taking 4th power of both,

$2^4\equiv2^4\pmod{17} \\ \\ 2^4\equiv16\pmod{17}$

Seems that 16 added to 1 is completely divisible by 17.

$16\equiv-1\pmod{17}$

Thus,

$2^4\equiv-1\pmod{17}$

So, taking the square...

$(2^4)^2\equiv(-1)^2\pmod{17} \\ \\ 2^8\equiv1\pmod{17}$

Now, take the 4th power.

$(2^8)^4\equiv1^4\pmod{17} \\ \\ 2^{32}\equiv1\pmod{17}$

Then, multiply 2 to both.

$2^{32} \times 2\equiv1 \times 2\pmod{17} \\ \\ 2^{33}\equiv \bold{2}\pmod{17}$

So 2 is the remainder.

Thus option 2 is the answer.

Answer 2

Answer:

2 is the required remainder....