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Question 307268: What is the units digit of 3^35?
a 1 b 3 c 5 d 7 e 9
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! The units digit repeats every 4 times.
3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561
3^9 = 19683
3^10 = 59049
3^11 = 177147
3^12 = 531441
3^13 = 1594323
3^14 = 4782969
3^15 = 1438907
3^16 = 43046721
3^17 = 129140163
3^18 = 387420489
3^19 = 1162261467
If you divide the exponent by 4, then the remainder will tell you what the equivalent base exponent is.
The base exponents are 0,1,2,3.
Using the base exponents, you get:
3^0 = 1 which gives you a units digit of 1.
3^1 = 3 which gives you a units digit of 3.
3^2 = 9 which gives you a units digit of 9.
3^3 = 27 which gives you a units digit of 7.
We'll run some tests with numbers we can verify and then extrapolate from that to numbers we can't verify easily.
We'll start with exponent of 16 and work up to 19.
16/4 = 4 with a remainder of 0.
3^0 = 1
3^16 = 43046721.
Units digit is confirmed to be equal to 1.
17/4 = 4 with a remainder of 1.
3^1 = 3
3^17 = 129140163
Units digit is confirmed to be equal to 3.
18/4 = 4 with a remainder of 2.
3^2 = 9
3^18 = 387420489
Units digit is confirmed to be equal to 9.
19/4 = 4 with a remainder of 3.
3^3 = 27
3^19 = 1162261467
Units digit is confirmed to be equal to 7.
Looks like our formula is good so we use it to find the units digit of a number we can't verify so easily.
We will now find the units digit for 3^35 using our formula.
35/4 = 8 with a remainder of 3.
Our base exponent is 3.
3^3 = 27
The units digit for 3^35 equals 7.
That would be selection d.
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