navin_c
Par 100 posts (V.I.P)
Number System- Lesson 4
CYCLICITY
Take any two numbers say 39 & 47.
If they are multiplied, the last digit of the product is same as the last digit of 9 x 7.
Hence, it is 3.
This concept could be extended to a host of situations. An interesting pattern emerges when we look at the exponents of the numbers. We would find conclusions as given below.
The last digits of the exponents of all numbers have cyclicity i.e. every Nth power of the base shall have the same last digit, if N is the cyclicity of the number. All numbers ending with 2, 3, 7, 8 have a cyclicity of 4.
For instance,
2^1 ends with 2
2^2 ends with 4
2^3 ends with 8
2^4 ends with 6
2^5 end with 2 again.
The same set of the last digits shall be repeated for the subsequent powers. So, if we want to find the last digit of (say) 2^45, divide 45 by 4.
The remainder is 1
So the last digit would be the same as last digit of 2^1, which is 2
Working out similarly for all other digits we get
DIGIT CYCLICITY
0, 1, 5 & 6 1
2, 3, 7 & 8 4
4 & 9 2
Walking on the given foot steps try out the following examples:
Find out the last digit of
1) 3^57
2) 7^23 x 8^13
3) 235^1000
CYCLICITY
Take any two numbers say 39 & 47.
If they are multiplied, the last digit of the product is same as the last digit of 9 x 7.
Hence, it is 3.
This concept could be extended to a host of situations. An interesting pattern emerges when we look at the exponents of the numbers. We would find conclusions as given below.
The last digits of the exponents of all numbers have cyclicity i.e. every Nth power of the base shall have the same last digit, if N is the cyclicity of the number. All numbers ending with 2, 3, 7, 8 have a cyclicity of 4.
For instance,
2^1 ends with 2
2^2 ends with 4
2^3 ends with 8
2^4 ends with 6
2^5 end with 2 again.
The same set of the last digits shall be repeated for the subsequent powers. So, if we want to find the last digit of (say) 2^45, divide 45 by 4.
The remainder is 1
So the last digit would be the same as last digit of 2^1, which is 2
Working out similarly for all other digits we get
DIGIT CYCLICITY
0, 1, 5 & 6 1
2, 3, 7 & 8 4
4 & 9 2
Walking on the given foot steps try out the following examples:
Find out the last digit of
1) 3^57
2) 7^23 x 8^13
3) 235^1000
it could be of help sometime....