In this section, you will learn how to find remainder when 2 power 256 is divided by 17.
Let us take exponents 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ....one by one for '2'.
For example, if we take exponent 3 for '2', we get
2^{3} = 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 the dividend itself has to be considered as 'Remainder'.
So, when 2^{3} is divided by 17, the remainder is 8.
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.
2^{n} 2⁰ 2^{1} 2^{2} 2^{3} 2^{4} 2^{5} 2^{6} 2^{7} 2^{8} 2^{9} |
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, 2^{0} is divided by 17 and the remainder is 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 of 2 is a multiple of 8.
In 2^{256}, the exponent is 256.
This is a multiple of 8. Because
256 = 8 x 32
Because 256 is a multiple of 8, when we divide 2^{256} 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 2^{263} is divided by 17
The exponent 263 is not a multiple of 8.
Write 263 in terms of multiple of 8 using division algorithm.
263 = 8 x 32 + 7
or
263 = 256 + 7
Then,
2^{263} = 2^{256 + 7}
2^{263 }= 2^{256 }x 2^{7}
In 2^{256}, the exponent 256 is a multiple of 8.
As explained in the above picture, using division algorithm, we can write 2^{256} as
2^{256} = 17k + 1
Then,
2^{263 } = 2^{256} x 2^{7}
2^{263 }= (17k + 1) x 2^{7}
2^{263 }= (17k + 1) x 128
Now we can get the remainder when 2^{263} divided by 17 as given below.
Therefore, the remainder is 9 when 2^{263} is divided by 17.
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