24 * 12 = 288
Note here: …4 * …2 = …8
So when we are looking at the units digit of the result of an integer raised to a certain exponent, all we need to worry about is the units digit of the integer.
Let’s look at the pattern when the units digit of a number is 2.
Units digit 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
…
Note the units digits. Do you see a pattern? 2, 4, 8, 6, 2, 4, 8, 6, 2, 4 … and so on
So what will 2^11 end with? The pattern tells us that two full cycles of 2-4-8-6 will take us to 2^8, and then a new cycle starts at 2^9.
2-4-8-6
2-4-8-6
2-4
The next digit in the pattern will be 8, which will belong to 2^11.
In fact, any integer that ends with 2 and is raised to the power 11 will end in 8 because the last digit will depend only on the last digit of the base.
So 652^(11) will end in 8,1896782^(11) will end in 8, and so on…
A similar pattern exists for all units digits. Let’s find out what the pattern is for the rest of the 9 digits.
Units digit 3:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
The pattern here is 3, 9, 7, 1, 3, 9, 7, 1, and so on…
Units digit 4:
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
The pattern here is 4, 6, 4, 6, 4, 6, and so on…
Integers ending in digits 0, 1, 5 or 6 have the same units digit (0, 1, 5 or 6 respectively), whatever the positive integer exponent. That is:
1545^23 = ……..5
1650^19 = ……..0
161^28 = ………1
Hope you get the point.
Units digit 7:
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = ….1 (Just multiply the last digit of 343 i.e. 3 by another 7 and you get 21 and hence 1 as the units digit)
7^5 = ….7 (Now multiply 1 from above by 7 to get 7 as the units digit)
7^6 = ….9
The pattern here is 7, 9, 3, 1, 7, 9, 3, 1, and so on…
Units digit 8:
8^1 = 8
8^2 = 64
8^3 = …2
8^4 = …6
8^5 = …8
8^6 = …4
The pattern here is 8, 4, 2, 6, 8, 4, 2, 6, and so on…
Units digit 9:
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = …1
The pattern here is 9, 1, 9, 1, 9, 1, and so on…
Summing it all up:
1) Digits 2, 3, 7 and 8 have a cyclicity of 4; i.e. the units digit repeats itself every 4 digits.
Cyclicity of 2: 2, 4, 8, 6
Cyclicity of 3: 3, 9, 7, 1
Cyclicity of 7: 7, 9, 3, 1
Cyclicity of 8: 8, 4, 2, 6
2) Digits 4 and 9 have a cyclicity of 2; i.e. the units digit repeats itself every 2 digits.
Cyclicity of 4: 4, 6
Cyclicity of 9: 9, 1
3) Digits 0, 1, 5 and 6 have a cyclicity of 1.
Cyclicity of 0: 0
Cyclicity of 1: 1
Cyclicity of 5: 5
Cyclicity of 6: 6
