find the remainder 2^51 is divided by 5
Answers
Answer:
Step-by-step explanation:
Well, here’s the easiest way I can think of:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
We notice that the unit’s place of EVERY FOURTH NUMBER repeats. So we imply from this that the CYCLICITY of the number 2 is FOUR.
Okay, getting back to 2^(51) divided by 5.
First off, we take the power, i.e. 51, and divide it by the cyclicity of the base number, i.e. 2 in this case.
=>51/4 gives a remainder of 3. So now, we take the remainder obtained on division and place that as the power.
=>2^3/5=8/5−−−> gives a remainder of 3, which is the required answer.
Given:
The number 2^51 is divided by 5.
To Find:
The remainder when 2^51 is divided by 5 is?
Solution:
1. The given exponential number is 2^51.
2. For a number to be divisible by 5, the units digit of the number must be either 0 or 5.
3. The remainder of a number when it is divided by 5 is,
- If the units digit is less than 5, the units digit is the remainder.
- If the units digit lies between 5 and 9 (both inclusive), then (Units digit - 5) is the remainder.
4. The units digit of 2^51 is calculated as,
=> The units digit of 2^n is,
- 2 for 4n + 1 type values,
- 4 for 4n + 2 type values,
- 8 for 4n + 3 type values,
- 6 for 4n type values.
=> 51 = 4(12) + 3 = 4n + 3 type values,
=> The units digit of 2^51 is 8.
5. The remainder when the units digit 8 is divided by 5 is 8-5 = 3.
Therefore, the remainder when 2^51 is divided by 5 is 3.