Question 1206604
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Since we are only looking for the units digit of the expression, we are only interested in the units digits of any calculations.  In other words, we are doing our calculations mod 10.<br>
13^35: The repeated pattern for successive powers of 3 is (3, 9, 7, 1). 35 divided by 4 leaves remainder 3, so the units digit of 13^35 is the 3rd number in the pattern, which is 7.<br>
57^30: The repeated pattern for successive powers of 7 is (7, 9, 3, 1). 30 divided by 4 leaves remainder 2, so the units digit of 57^30 is the 2nd number in the pattern, which is 9.<br>
34^33: The repeated pattern for successive powers of 4 is (4, 6). 33 divided by 2 leaves remainder 1, so the units digit of 34^33 is the 1st number in the pattern, which is 4.<br>
Then the units digit of the expression is the units digit of 7+9+4=20, which is 0.<br>
ANSWER: 0<br>
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For other tutors who might be looking at this problem....<br>
When I first tried to post a response to this problem, some weird stuff happened in the middle of writing my response, and I got kicked out of the site.  Possibly whatever happened to cause that caused the statement of the problem to be corrupted.<br>
As I hope you can see now, the expression for which we were to find the units digit is<br>
13^35 + 57^30 + 34^33<br>