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Ten's Digit

There lies the cyclicity of tens’ place digit of all the digits. This is given below:
 
Digits Cyclicity
2, 3, 8– 20
4, 9 – 10
5 1
6 5
7 4
 
Example
What is the tens’ place digit of 1242?
Solution
For this, we need to break 1242 first by using binomial theorem as (10 + 2)42. Obviously this expression will have 43 terms, and out of these 43 terms first 41 terms will have both of their tens and units place digit as 0.
Last two terms will be → 42C41 × 101 × 241 + 42 C42 × 100 × 242
Now, we will find the tens’ place digit of all these terms individually.
Tens digit of 42C41 × 101 × 241 = 42 × 10 × (02) [Cyclicity of 2 is 20, so 241 will have same tens digits as 21] = 840, so 40 are the last two digits.
Similarly, 42C42 × 100 × 242 = 1 × 1 × 04 = 04
So, finally last two digits are → 40 + 04 = 44, so 4 is the tens’ place digit.
 
 
Note: (25)n and (76)n will always give 25 and 76 as the units and tens digit for any natural number value of n.




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