Find the last digit of the sum of two numbers with large exponents.

johnnyICON

Hi, I am trying to solve this question in my textbook. It asks me to find the last digit of the following expression:

$$222^{555} + 555^{222}$$

Now, I have tried using modular arithmetic to solve this question, but I'm stuck.
And I'm not all too sure if took the right approach or not, any hints?

shmoe

Homework Helper
You should be able to do the two terms seperately, can you find 555^222 mod 100? 222^555 mod 100?

It might make the arithmetic a little easier if you consider powers of 2, 5, and 11 mod 100 seperately but this isn't mandatory.

johnnyICON

I did something like that. I used mod 11 for both numbers. Found 222 congruent to 2 mod 11, and 555 congruent to 5 mod 11.

So I made the statement that $$222^{555} \equiv 2^{555} mod 11$$and $$555^{222} \equiv 5^{222} mod 11$$

Then, I did successive powers of both 2 and 5 and found their congruencies until I found a pattern.

I determined that after five successive powers of 5, the congruencies would repeat. And after ten successive powers of 2, the congruencies would repeat.

So I divided the the exponents 222 and 555 by 10 and 5, respectively, to find another congruency. And that's where I get stuck.

shmoe

Homework Helper
Sorry, I was misleading in my last post. For some reason I thought you wanted the last two digits, not just the last one. To find the last digit you need only find 222^555 and 555^222 mod 10, which will be easier than finding them mod 100. This also means the bit about 11 is superfluous.

johnnyICON

Maybe you can take a look at this example:
Example. Find the last digit of $$7^{100}$$.
We need to evaluate $$7^{100}$$ (mod 10). Since $$7^2$$ = 49$$\equiv$$ −1 (mod 10) we have $$7^{100} \equiv (72)^{50} \equiv$$ (−1)$$^{50} \equiv 1$$ (mod 10).Thus the last digit is 1.
Is the last digit 1 because of the last congruency?

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shmoe

Homework Helper
Yes, the last digit of 7^100 is 1 because they've shown that the remainder you get when you divide 7^100 by 10 is 1.

So you look at the powers of 2 and find the pattern is 4 digits long. So 2^1, 2^5, 2^9, 2^13, ...2^105, ... are all congruent to 2 mod 10. Essentially you can add a 4 to the exponent and not affect the result. Same with 2^2, 2^6,... In the language of congruences, if $$a\equiv b\ mod\ 4$$ then $$2^a\equiv 2^b\ mod\ 10$$. There is a good reason why this is true, a power of 2's congruence class mod 10 is completely determined by its congruence class mod 5 (Chinese remainder theorem if you've met it) and we know $$2^4\equiv 1\ mod\ 5$$.

In the link you provided, they have $$a\equiv b\ mod\ 4$$ implies $$7^a\equiv 7^b\ mod\ 10$$ (we actually know here $$7^4\equiv 1\ mod\ 10$$). So to find $$7^{358}\ mod\ 10$$ notice $$358\equiv 2\ mod\ 4$$ so $$7^{358}\equiv 7^2\equiv 9\ mod\ 10$$.

johnnyICON

Wow, that makes a lot of sense now. And thanks for relating to the link I provided. I was really uncertain about this part::
shmoe said:
notice $$358\equiv 2\ mod\ 4$$ so $$7^{358}\equiv 7^2$$
When I divided the exponent by the length and found its congruence, I wasn't sure if it meant that $$7^{358}\equiv 7^2$$ or if it meant $$7^{358}\equiv 7^{2 (mod 4)}$$.

So when dealing with congruencies, I can replace a with b by symmetry right? That is, $$358\equiv 2\ mod\ 4$$ is equivalent to $$2\equiv 358\ mod\ 4$$.

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shmoe

Homework Helper
johnnyICON said:
When I divided the exponent by the length and found its congruence, I wasn't sure if it meant that $$7^{358}\equiv 7^2$$ or if it meant $$7^{358}\equiv 7^{2 (mod 4)}$$.
$$7^{358}\equiv 7^2\ mod\ 10$$ would be the correct one. You don't usually put "mod" up in an exponent.

johnnyICON said:
So when dealing with congruencies, I can replace a with b by symmetry right? That is, $$358\equiv 2\ mod\ 4$$ is equivalent to $$2\equiv 358\ mod\ 4$$.
Yep. The "modular equivalence" works in many ways like your usual equal sign. It's symmetric, transitive and reflexive.

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johnnyICON

Great! Thanks a lot shmoe, you've helped a lot! Too bad I couldn't rate your help.
10/10

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