Try doing 3 to some easy numbers and look for a pattern.
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
Notice the pattern of last digits goes 3, 9, 7, 1 and repeats with a period of 4.
46 divided by 4 gives a remainder of 2 so the answer is the same as 3^2 which is 9.
it is 3
3 to a power divisible by 4 will have a units digit of 1.The powers of 3 are 3, 9, 27, 81 ... obviously, the next one will have a units digit of 1x3 or 3, the next one will have a units digit of 3x3 or 9, the next one will have a units digit of 7 (because 9x3 is 27), the next one will have a units digit of 1 (because 7x3 is 21), and then the cycle starts over with a units digit of 3 again.
The units digit of 20132013 is the same as the units digit of 32013. The units digit of 34 = units digit of 81 = 1 So units digit of 32013 = 32012+1 = 34*503+1 = 34*503 *31 = 1503*3 = 3
7
Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.
Power 2: units digit 9. Multiply by 49 again to get power 4: units digit 1. So every 4th power gives units digit 1. So 16th power has units digit 1, so the previous power, the 15th must have units digit 3.
3
it is 3
3 to a power divisible by 4 will have a units digit of 1.The powers of 3 are 3, 9, 27, 81 ... obviously, the next one will have a units digit of 1x3 or 3, the next one will have a units digit of 3x3 or 9, the next one will have a units digit of 7 (because 9x3 is 27), the next one will have a units digit of 1 (because 7x3 is 21), and then the cycle starts over with a units digit of 3 again.
The units digit of 20132013 is the same as the units digit of 32013. The units digit of 34 = units digit of 81 = 1 So units digit of 32013 = 32012+1 = 34*503+1 = 34*503 *31 = 1503*3 = 3
1
7
Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.Since neither the three hundred, nor the ten can contribute to the units digit in the answer, you look for a pattern in the units digit in the powers of 2n.20 = 121 = 222 = 423 = 824 = 2and after that , the pattern repeats, 4, 8, 2, 4, 8, 2, ...So if n (mod 3) = 1 the units digit is 2if n (mod 3) = 2 the units digit is 4and if n (mod 3) = 0 the units digit is 8where n (mod 3) is the remainder when n is divided by 3.312 is divisible by 3 [3+1+2=6 is divisible by 3] so 312 mod(3) =0 and so the units digit is 8.
I guess you mean what's the units digit of 32011. It is 7. To work this out, see how the units digit of 3n changes; it goes: 3, 9, 7, 1, 3, 9, 7, 1, ... (only the first 8 powers are shown) repeating the same sequence of 4 digits. So if we find the remainder of 2011 divided by 4, it will tell us which of the four numbers (3, 9, 7, 1) will be the units digit of 32011: 2011 ÷ 4 ⇒ remainder 3, so the 3rd digit is the required digit: 7. (If there had been no remainder, then the 4th digit, namely 1, would have been the required value.)
3
6308 6321
1ExplanationNote the following:31=332=933=2734=8135=24336=72937=218738=6561You will see that the units digit of the powers of three cycles through the following: 3, 9, 7, 1. 3 raised to a multiple of 4 has 1 as the units digit. Because 2008 is a multiple of 4, 32008 has 1 in the units digit (and is thus the remainder when that number is divided by 5).