Concept: Unit's Digit
- Units's digit of sum of two numbers same as unit's digit of sum of their respective unit's digits.
- Units's digit of product of two numbers same as unit's digit of product of their respective unit's digits.
Example: Units digit of 126 + 589 = Units digit of 6 + 9 = 5
Example: Units digit of 126 × 589 = Units digit of 6 × 9 = 4
Let us calculate the unit's digit of various powers of 2.
Unit's digit of 21 = 2
Unit's digit of 22 = 4
Unit's digit of 23 = 8
Unit's digit of 24 = 6
Unit's digit of 25 = 2
Unit's digit of 26 = 4
Unit's digit of 27 = 6
Unit's digit of 28 = 8
Unit's digit of 29 = 2
...
∴ We see that the unit's digit of powers of 2 repeates after every 4 powers. Hence, we can say that unit's digit of powers of 2 has a cyclicity of 4 terms.
Unit's digit of 24k = 6
Unit's digit of 24k+1 = 2
Unit's digit of 24k+2 = 4
Unit's digit of 24k+3 = 8
- 2 has a cyclicity of 4 terms.
- 3 has a cyclicity of 4 terms.
- 4 has a cyclicity of 2 terms.
- 5 has a cyclicity of 1 term.
- 6 has a cyclicity of 1 term.
- 7 has a cyclicity of 4 terms.
- 8 has a cyclicity of 4 terms.
- 9 has a cyclicity of 2 terms.
- 0 has a cyclicity of 1 term.
Unit's digit of powers of:
In general, we can say unit's digit of power of any number has a cyclicity of 4.
- Unit's digit of
- 4odd number = 4
- 4even number = 6
- Unit's digit of
- 9odd number = 9
- 9even number = 1
- Unit's digit of a4k+n = Unit's digit of an
where a is any single digit number. - Units's digit (...abc)x = Units's digit of cx.
Example: Units digit of 12654 = Unit's digit of 654 = 6 - Unit's digit of (5 × odd number) = 5
- Unit's digit of (5 × even number) = 0
Example: Find unit's digit of (123)23
Solution:
Unit's digit of (123)23 = Unit's digit of 323
= Unit's digit of 34×5 + 3
= Unit's digit of 33
= 7
Example: Find unit's digit of (23454)769
Solution:
Unit's digit of (23454)769 = Unit's digit of 4769
We know Unit's digit of 4odd number = 4
∴ Unit's digit of (23454)769 = 4
Example: Find unit's digit of (582)945
Solution:
Unit's digit of (582)945 = Unit's digit of 2945
= Unit's digit of 24×236 + 1
= Unit's digit of 21
= 2
Example: Find unit's digit of
Solution:
Unit's digit of = Unit's digit of
Now, 6125 when divided by 4 leaves a remainder of 1 [Concept of Remainders]
∴ Unit's digit of = Unit's digit of 74k + 1
= Unit's digit of 71
= 7