Cyclicity

1. CYCLICITY OF UNIT DIGIT

2. • The concept of cyclicity is used to identify the unit digit of any
power .
• The unit digit in any operation depends on only the unit digit of
the numbers used .

3. • Find the unit digit of 357 4084
• To find the unit digit ,we do not have to perform the complete
multiplication .The unit digit will just depend on the unit digits
of the two numbers ……………….

4. •
• 3 5 7
• 4 0 8 4
• ? ? ? ?

5. We
don’t need
•
to calculate
whole for
unit digit
• 3 5 7
• 4 0 8 4
• ? ? ? ?
8

6. We would usually have to find the unit d igit of a power e.q. unit digit of
19 51 19
4132 or 27 or so on . A s learnt above the unit digit of 4132
51 19 31
or of 27 w ill be respectively the sam e a s the unit digit of 2 and that of 7
Thus we have to acquaint ourselves with the unit digit of powers of 1 to 9 .
The unit digit of successive powers of any number repeat themselves in a particular cycle

7. • Concept of a cycle & finding any term of a cycle
• Finding the unit digit of successive powers of 2 ……..
•

8. • Powers of 2
1 2 3 4 5 6 7 8
2 2 2 2 2 2 2 2
• Unit Digit
•
•
2 4 6 8 2 4 6 8
•

9. • Once we know the cycle, any term in the cycle can be found out.
• We see that the cycle ends at every multiple of 4 and then cycle
starts again . To generalize ,if the power of 2 is one more than
multiple of 4 i.e. 4a + 1 then unit’s digit will be 2 ,if the power is
two more than a multiple of 4 i.e. 4a + 2 , then the unit’s digit
will be 4 ; so on

10. • Cyclicity of 1 ,5 and 6
• Any power of 1 ,5 and 6 will end respectively with 1 , 5 and 6 (
irrespective of the power ) . Thus the digits 1 , 5 and 6 are said
to have a cyclicity of 1 and finding the unit digit of their power is
not an issue

11. • Cyclicity of 4 and 9
• The digit 4 and 9 have a cyclicity of 2 i.e. two different values of
the unit digit get repeated ,as the powers of 4 and 9 are
increased successively.
n
C yclicity of unit digit of 4 as n takes values 1 ,2 ,3 ... is 4 , 6 , 4 ,
n
C yclicity of unit digit of 9 as n takes values 1 ,2 ,3 ... is 9 , 1, 9 ,
• Thus in the case of 4 and 9 one just needs to check if n is odd or
even and accordingly the unit digit can be found

12. • Cyclicity of 2 ,3 , 7 and 8
• Each of these digits has a cycle of 4 and the respective cycles
are …..
As n assumes successive natural numbers 1 ,2 ,3 4 …………

13. •
n
U nit digit of 2 w ill be 2 ,4 , 8 , 6
n
U nit digit of 3 w ill be 3 , 9 , 7, 1
n
U nit digit of 7 w ill be 7 , 9 , 3 , 1
n
U nit digit of 8 w ill be 8 ,4 , 2 , 6