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Cyclicity of unit digit

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Cyclicity

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Cyclicity of unit digit

1. 1. CYCLICITY OF UNIT DIGIT
2. 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. 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. 4. •• 3 5 7• 4 0 8 4• ? ? ? ?
5. 5. We don’t need• to calculate whole for unit digit• 3 5 7• 4 0 8 4• ? ? ? ? 8
6. 6. We would usually have to find the unit d igit of a power e.q. unit digit of 19 51 194132 or 27 or so on . A s learnt above the unit digit of 4132 51 19 31or of 27 w ill be respectively the sam e a s the unit digit of 2 and that of 7Thus 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. 7. • Concept of a cycle & finding any term of a cycle• Finding the unit digit of successive powers of 2 ……..•
8. 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. 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. 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. 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. 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. 13. • nU 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