To find the units digit of a large number?

One final question here is another example

If we have the find the units digit of \(344^{328}\)

\(4^1\) is 4
\(4^2\) is 16
\(4^3\) is 4

so the repeating block over here {4,6}

In this case the remainder is 0 so the units digit of this expression is 6?

so if there are 4 repeating blocks and the remainder is 0 we raise it to the 4th power ( some examples would be 3,7 etc) in this current example its the 2nd power?

Yes, if the remainder is 0, then take the last digit from the block.

fozzy,

you are correct in both cases. here is how i like to think about it:
7^{777} example:
If we divide 777 by 4, the quotient is 194 and the remainder is 1.
This means that we will have 194 of {7, 9, 3, 1} repeating blocks, and we will have 1 more term left, and the units digit of 7^{777} will be 7, the first term in the repeating block.

When we look at the case of 344^{328}, the units digit of powers of 4 cycle as {4, 6}, when we divide 328 by 2, the remainder is 0, this means that there will be exactly 164 blocks consisting of {4,6} without any remainder and the units digit of 344^{328} will be 6.

here are problems using the same concept(some hard):
#1) if-n-is-a-positive-integer-what-is-the-remainder-when-82380.html
#2) what-is-the-remainder-when-7-345-7-11-2-is-divided-by-26794.html
#3) remainder-when-7-4n-3-6-n-104848.html
#4) if-3-4n-1-is-divided-by-10-can-the-remainder-be-0-a-4662.html
#5) what-s-the-remainder-of-2-x-divided-by-10-1-x-is-an-even-9651.html

cheers,
dabral

Finding the UNIT’S DIGIT - please share some more useful tricks if its there.

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