# Ten's Digit

There lies the cyclicity of tensâ€™ place digit of all the digits. This is given below:

Digits |
Cyclicity |

2, 3, 8â€“ | 20 |

4, 9 â€“ | 10 |

5âˆ’ | 1 |

6âˆ’ | 5 |

7âˆ’ | 4 |

Example

What is the tensâ€™ place digit of 12

^{42}?Solution

For this, we need to break 12

^{42}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 â†’

^{42}C_{41 }Ã— 10^{1 }Ã— 2^{41}+^{42 }C_{42 }Ã— 10^{0 }Ã— 2^{42}Now, we will find the tensâ€™ place digit of all these terms individually.

Tens digit of

^{42}C_{41 }Ã— 10^{1 }Ã— 2^{41 }= 42 Ã— 10 Ã— (02) [Cyclicity of 2 is 20, so 2^{41}will have same tens digits as 2^{1}] = 840, so 40 are the last two digits.Similarly,

^{42}C_{42 }Ã— 10^{0 }Ã— 2^{42 }= 1 Ã— 1 Ã— 04 = 04So, finally last two digits are â†’ 40 + 04 = 44, so 4 is the tensâ€™ place digit.

**Note:**(25)

*and (76)*

^{n}*will always give 25 and 76 as the units and tens digit for any natural number value of*

^{n}*n*.