Unit Digit of Power of Positive Integer a^n
| U(a) | U(a^2) | U(a^3) | U(a^4) | U(a^5) | U(a^6) | U(a^7) | U(a^8) | U(a^9) |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 16 | 32 | 64 | 128 | 512 | 1024 |
| 3 | 9 | 27 | 81 | 243 | 729 | 2187 | 6561 | 19683 |
| 4 | 16 | 64 | 256 | 1024 | 4096 | 16384 | 65536 | 262144 |
| 5 | 25 | 125 | 625 | 3125 | 15625 | 78125 | 390625 | 1953125 |
| 6 | 36 | 216 | 1296 | 7776 | 46656 | 279936 | 1679616 | 10077696 |
| 7 | 49 | 343 | 2401 | 16807 | 117649 | 823543 | 5764801 | 40353607 |
| 8 | 64 | 512 | 4096 | 32768 | 262144 | 2097152 | 16777216 | 134217728 |
| 9 | 81 | 729 | 6561 | 59049 | 531441 | 4782969 | 43046721 | 387420489 |
Unit Digit Cyclicity
- 1,5 & 6 – The unit digit of their power does not change. i.e
a^n = a
- 2,3,7 & 8 – The unit digit of their power is getting repeated after every 4th cycle
- 2 – The unit digit is 6 when the power is 4n
- 2 – The unit digit is 8 when the power 4n+3
- 2 – The unit digit is 4 when the power 4n+2
- 2 – The unit digit is 2 when the power 4n+1
- 3 – The unit digit is 1 when the power 4n
- 3 – The unit digit is 7 when the power 4n+3
- 3 – The unit digit is 9 when the power 4n+2
- 3 – The unit digit is 3 when the power 4n+1
- 7 – The unit digit is 1 when the power 4n
- 7 – The unit digit is 3 when the power 4n+3
- 7 – The unit digit is 9 when the power 4n+2
- 7 – The unit digit is 7 when the power 4n+1
- Same for 7 & 8
- 4 & 9 – The unit digit repeat after every 2 cycles
- 4 – The unit digit is 4 for every odd power i.e 2n+1
- 4 – The unit digit is 6 for every even power i.e 2n
- 9 – The unit digit is 9 for every odd power i.e 2n+1
- 9 – The unit digit is 1 for every even power i.e 2n
Last Two Digits Cyclicity
- The last two digits of 5^n (n>=2) is 25
- The ordered pair of last two digits of 6^n (n>=2) changes with the period – 36, 16,96,76,56 as n changes. The repetition starts from power of 2.
- 6^5n+1 ends with 56
- 6^5n+2 ends with 36
- 6^5n+3 ends with 16
- 6^5n+4 ends with 96
- 6^5n ends with 76
- 7^4n+1 ends with 07
- 7^4n+2 ends with 49
- 7^4n+3 ends with 43
- 7^4n ends with 01
- The ordered pair of last two digits of 7^n (n>=1) changes with the period – 07,49,43,01 as n changes.
- The ordered pair of last two digits of 76^n is always 76
Last two digits of
= last two digits of 
= last two digits of 
Q. What is the unit digit for 7^200?
A. 200 = 4n Therefore it will be 1
Q. What is the unit digit for 8687^200?
A. The unit digit of 8687 is 7. And we know 7^200 the power is 4n Therefore it will be 1
Q Unit digit of 7^61 + 9^32?
A. For 7^61 – unit digit is 7
For 9^32 – unit digit is 1
Therefor 7+1 = 8
Q. Unit digit of 5^61 x 9^32 x 7^2013?
A. For 5^61 unit digit – 5
For 9^32 unit digit – 1
For 7^2013 unit digit – 7
Therefor 5 x 1 x 7 = 35 i.e 5 in unit digit
Another approach – 9^32 is always odd. As any power of an odd number is always odd
7^2013 is always odd. Same reason
9^32 x 7^2013 – Multiplication of two odd number is always odd
5^61 x 9^32 x 7^2013 – Multiplication of 5 with an odd number, the unit digit is 5
Q. Unit digit of 4^2022^2023 + 9^2023^2022 ?
A. 2022^2023 – This will always be even. As any power of even number is even. The unit digit for the even power of 4 is 6
2023^2022 – This will always be odd. As any power of odd number is odd. The unit digit for the odd power of 9 is 9
Therefor 6 + 9 = 15 i.e the unit digit will be 5
Q. The last two digit of 6^202?
A. When the power of 6 is 5n+2 the last two digits are 36
| a | a^n | a^2n (Even) | a^2n+1 (odd) | a^4n | a^4n+3 | a^4n+2 | a^4n+1 |
|---|---|---|---|---|---|---|---|
| 1 | 1 | ||||||
| 5 | 5 | ||||||
| 6 | 6 | ||||||
| 4 | 6 | 4 | |||||
| 9 | 1 | 9 | |||||
| 2 | 6 | 8 | 4 | 2 | |||
| 3 | 1 | 7 | 9 | 3 | |||
| 7 | 1 | 3 | 9 | 7 | |||
| 8 | 6 | 2 | 4 | 8 |
