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How to prove that every power of 3 has ten's digit even.how to approach these kind of number theory questions.


marked as duplicate by Martin R, Ethan Bolker, peterwhy, Gerry Myerson elementary-number-theory Jun 9 at 13:12

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    $\begingroup$ Possible duplicate of In every power of 3 the tens digit is an even number (found in less than 30 seconds Googling :) $\endgroup$ – Martin R Jun 9 at 12:50
  • $\begingroup$ How to know that the question i am going to ask has already been asked by someone else. $\endgroup$ – Aryan 24k Jun 9 at 14:16
  • $\begingroup$ Whether you can provide me more theroetical answer then this approach. $\endgroup$ – Aryan 24k Jun 9 at 14:18

We know that $3^1=03$




And so on. Here we noticed that 81 is the largest two digit number that is in the form of $3^n$. After that 3 digit number starts . But till 81 if you see tens digit is even so here 2 case arises Case 1 When we have last digit of $3^n$ = 3, 1 After multiplying it by 3 we see it not makes any effect on the ten's digit and as ten's digit is even then any thing multiplied to even will give you even So for this case ten's digit is even.

Case 2 When the last digit of $3^n$ = 9 , 7 Here after multiplying by 3 it will have a effect on ten's digit But $9*3=27$ And$3*7=21$ Here wee see 2 will be added to ten's digit Now again as before I said ten's digit is even then the product after multiplying it by 3 will be even and as 2 is added to the result it will also be an even number as 2 is even and even added to even gives us even. Here's what I thought.


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