To find the units digit of a large number?

Question

So if we are given a question what is the units digit of  777^{777}

we find the pattern for 7 (7,9,3,1)

then we divide  \frac{777}{4}  and the remainder is 1 so the units digit is  7^1  which is 7?

Is this correct?

fameatop · Answer

Absolutely Correct.

Bunuel · Answer

Yes.

The units digit of 777^777 = the units digit of 7^777.

7^1 has the units digit of 7;
7^2 has the units digit of 9;
7^3 has the units digit of 3;
7^4 has the units digit of 1.
7^5 has the units digit of 7 AGAIN.

The units digit repeats in blocks of 4: {7, 9, 3, 1}...

The remainder of 777/4 is 1, thus the units digit would be the first number from the pattern, so 7.

Hope it's clear.

fozzzy · Answer

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?

Bunuel · Answer

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

dabral · Answer

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

MayankSatkar · Answer

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

bumpbot · Answer

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To find the units digit of a large number?

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