#1**0 **

The last 10 digits of 2^2010 =...........6606081024

Note: Somebody should give you a short-cut to this computation.

Guest Oct 13, 2020

#3**+1 **

Notice that 2^{n} for n = 1 • • • the value of 2^{n} is as follows:

2, 4, 8, 16,

32, 64, 128, 256,

512, 1024, 2048, 4096,

8192, 16384, 32768, 65536,

etc.

The units digit repeats after every four exponents.

Example: Say you wanted to know 2^{7} ... divide the 7 by 4 and the remainder is 3.

So the units digit will be the third one in the repeating groups, which is 8.

Example: Or, you wanted to know 2^{16} ... divide the 16 by 4 and the remainder is 0.

But when a remainder is 0, that's the same as it being the same as the divisor, which was 4.

So the units digit will be the fourth one in the repeating groups, which is 6.

Okay, skip ahead to 2^{2010} ... divide the 2010 by 4 and the remainder is 2.

That means the units digit will be the second one in the repeating groups, which is **4**.

_{.}

Guest Oct 13, 2020