What is the remainder when 2^20 is divided by 10 ?

A. 0
B. 2
C. 4
D. 6
E. 8

2^20 = 2^10 x 2^10

And 2^10 = 1024

4*4= 16, so the unit digit of this multiplication is 6

The remainder is 6

\(\frac{{2^{20}}}{10}\) = \(\frac{{2^{20}}}{2*5}\) = \(\frac{{2^{19}}}{5}\)

\(\frac{{2^1}}{5}\) = 2

\(\frac{{2^2}}{5} = 4\)

\(\frac{{2^3}}{5} = 3\)


\({2^{19}}\) = \({2^{3*6}}\) x \(2^1\)

\(\frac{{2^{3*6}}}{5}\) = Remainder 3

\(\frac{2^1}{5}\) = Remainder 2

So, Result will be 3*2 = 6 , Answer will be (D)

PS: Its better to avoid this approach during actual GMAT exam, just posting an alternate approach for educational purpose.
_________________

How come you can't just take 2^19/5 and say that it is remainder 3? Isn't 2^20/10 = 2^19/5? Yet you get a remainder of 6 for the first one and a 3 for the second. I'm confused how you knew to multiply the remainder of 2^18/5 and 2/5.

Easy explanation:

\(2^{20}=4^{10}=16^5\)

All powers of 6 end with a 6 in the units digit, so \(16^5\) must also end with a 6. Thus, when divided by 10, the remainder must be 6.

Sure! When you multiply 20 2s together, you get 10 4s because each pair of 2s makes a 4. (2 x 2 = 4) When you multiply 10 4s together, you get 5 16s because every pair of 4s makes a 16 (4 x 4 = 16).

Every power of 6 ends with a 6 because \((6)(6) =36\) and \((6)(6)(6) = (36)(6) = 216\) and \((6)(6)(6)(6) = 1296\), etc. And the remainder when you divide by 10 will always be equal to 6, because multiples of 10 always end in 0. \(36/10 = 3 R 6, 216/10 = 21 R 6, 1296/10 = 129 R 6\), etc.

We need to determine the remainder when 2^20 is divided by 10. To do so, recall that any number divided by 10 will produce the same remainder as the units digit of that number. Thus, let’s determine the units digit of 2^20. The pattern of units digits of 2 when raised to a positive integer exponent is:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 6

2^5 = 2

We see that the pattern is 2-4-8-6. Furthermore, 2^4k, in which k is a positive integer, will always have a units digit of 6.

Thus, the units digit of 2^20 is 6. Dividing 6 by 10 yields a remainder of 6; thus, dividing 2^20 by 10 also yields a remainder of 6.

Answer: D
_________________

For any given number, when divide by 10, the last digit is basically the remainder of the operation.
We have to find the last digit of 2^20.

Cyclicity of 2 is also 4:

2^1-> 2
2^2-> 4
2^3-> 8
2^4->6
2^5->2...

Here 2^20 is basically (2^5)^4 so last digit 6.

Yes correct, just rewriting the highlighted part for you: (2^10)^2.

2^10^2 might mean 2^100. Cheers.

amegupte0410

Asked: What is the remainder when 2^20 is divided by 10 ?

2^20mod10 = 2^{5*4}mod10 = 32^4mod10 = 2^4mod10 = 16mod10 = 6

IMO D
_________________