"What is the remainder when 2 power 256 is divided by 17?" is a bit challenging question for the people who are getting prepared for competitive exams.
Actually it is not a difficult question and it can be answered easily once we know the stuff.
Let us take exponents 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ....one by one to "2"
For example, if we take exponent "3" for 2, we get 2³ = 8
Here 8 is less than the divisor 17 and 8 can not be divided by 17.
If the dividend is less than the divisor, then that dividend itself to be considered as "Remainder"
If the dividend is greater than the divisor, then we have to divide the dividend by the divisor and get remainder.
Let us deal our problem in this way.
2⁰ 2¹ 2² 2³ 2⁴ 2⁵ 2⁶ 2⁷ 2⁸ 2⁹ |
Value 1 2 4 8 16 32 64 128 256 512 |
Remainder 1 2 4 8 16 15 13 9 1 2 |
When we look at the above table carefully, 2⁰ is divided by 17, we get the remainder "1".
Again we get remainder "1" for power "8".
Next we will get remainder "1" for power "16.
So, we get remainder "1", if the exponent is a multiple of "8".
In 2²⁵⁶, the exponent is 256.
This is a multiple of 8. Because 256 = 8 x 32.
Since the exponent 256 is a multiple of 8, when we divide 2²⁵⁶ by 17, the remainder will be 1.
Let assume an exponent which is not a multiple of "8".
For example, say the exponent is 263
What will be the remainder, if 2²⁶³ is divided by 17
Since the exponent is not a multiple of 8, split the given exponent in terms of multiple of "8".
Using division algorithm, we get 263 = 8x32 + 7
or 263 = 256 + 7
Then, 2²⁶³ = 2²⁵⁶⁺⁷
2²⁶³ = 2²⁵⁶ x 2⁷
In 2²⁵⁶, the exponent 256 is a multiple of 8.
As explained in the above picture, using division algorithm, we can write 2²⁵⁶ as
2²⁵⁶ = 17k + 1
Then, 2²⁶³ = 2²⁵⁶ x 2⁷ ---------------> 2²⁶³ = (17k + 1) x 2⁷
2²⁶³ = (17k + 1) x 2⁷ ---------------> 2²⁶³ = (17k + 1) x 128
Now we can get the remainder when 2²⁶³ divided by 17 as given below.
Hence, the remainder is " 9 " when 2²⁶³ is divided by 17.
After having gone through the stuff and examples explained, we hope that the students would have understood, how to get remainder when 2 power 256 is divided by 17
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