What is the remainder when 2^(power) 1000 is divided by 5?

Question

and please this is not my home work and please either give a an explaination only if u know please no guesses and do ur work on ur own answers

Answer 1

This can be done using modular arithmeteic

we know

2^4 = 16 = 1 mod 5
2^1000 = (2^4)^250
so 2^1000 mod 5 = (2^4)^ 250 mod 5
= (2^4 mod 5)^ 250 mod 5
= 1^250 mod 5
= 1
so remainder = 1

Answer 2

This question can easily be answered by observing the pattern

Lets see:-

=> (2^2)/5 gives remainder as 4
=> (2^4)/5 gives remainder as 1
=> (2^6)/5 gives remainder as 4
=> (2^8)/5 gives remainder as 1

When ever the power is a multiple of 2 and 4 then the remainder is 1 and not when if is only divisible by 2

As 1000 is divisible by 2 and 4 therefore the answer is 1 i.e. the remainder

Hence is could be answered by a mere observation

Answer 3

Powers of 2 will have a reminder when divided by 5 that follows this sequence as you go up in power: 1 2 4 3.
1 (2^0): 1
2 (2^1): 2
4 (2^2): 4
8 (2^3): 3
16 (2^4): 1
32 (2^5): 2
64 (2^6): 4
128 (2^7): 3
256 (2^8): 1
and so on.
Notice that with each exponent of two that is a multiple of 4 will then have a reminder of 1 when divided by 5?

By the time you make it to 2^1000, since that exponent is divisible by 4, you should thus have a reminder of 1.
So, that is your answer.

(What purpose can this have, by the way?)

Answer 4

2^1000 = 2^(2*500) = (2^2)^500 = 4^500.

Now, 2^1000 = 4^500 =(5-1)^500.
Use binomial formula to get:

= C(500,0) 5^500[(-1)^0] + C(499,1) 5^499(-1)^1 +...+
C(1,499) 5^1(-1)^499 +C(500,500) (5^0)(-1)^500.

Now, all the terms in the sum are divisible by 5 except the last which is 1*1*1=1, so the sum of these terms when divided by 5 gives a remainder of 1.

Answer 5

There is a pattern... 2, 4, 3, 1.

2^1=2 / 5 = R2
2^2=4 / 5 = R4
2^3=8 / 5 = 1 R3
2^4=16 / 5 = 3 R1
2^5=32 / 5 = 6 R2

Do you see the pattern?

Take the 1000 / 4 = 250 with no remainder... so we take the fourth pattern answer...

Remainder = 1

Let's check it with another power level (i.e. 16).

16 / 4 = 4 with no remainder, so we should be able to take the fourth pattern answer.

2^16 = 65,536 / 5 = 13,107 R1

Let's try one more just to be safe... 2^21.

21 / 4 = 5 R1

So, we should be able to take the first pattern answer - let's check it out! Remainder should be 2.

2^21 = 2,097,152 / 5 = 419,430 R2

There we go!

Answer 6

math_kp's solution is elegant, simple, and correct.

Answer 7

Why should i tell you? Is this for homework?......and the answer is reemainder 4

Answer 8

2.14301^300