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Unit Digit

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by patelv » Sun Aug 29, 2010 11:08 pm
What's the unit digit of 365^365+364^364+363^363+362^362+361^361?
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by uwhusky » Mon Aug 30, 2010 12:30 am
You can figure out the answer by calculating the recurring pattern of each number with its exponent, and then add the unit digit together.

For example, 5^any positive number will always result in unit digit 5.

4^2 = unit digit 6, 4^3 = unit digit 4, and it repeats from there. So 364^364 must result in unit digit of 6.

Do the same with rest of the numbers and you got your answer.
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by patelv » Mon Aug 30, 2010 1:00 am
Thanks dwhusky for your reply.My solution is as follows:

(Can you pls tell me is this correct?)

The Unit digit for 365^365 is 5
The Unit digit for 364^364 is 6
The Unit digit for 363^363 is 7
Coz the repetative Unit digit for 3 is 3,9,27,81........
So the Unit digit of 363rd digit will be 7
The Unit digit for 362^362 is 4
Coz the repetative Unit digit for 2 is 2,4,8,16,........
So the Unit digit of 362nd digit will be 4
The Unit digit for 361^361 is 1.

So Addition of all the Unit Digit i.e 5+6+7+4+1 =23
So Ans is 3 (Unit Digit)
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by Rahul@gurome » Mon Aug 30, 2010 1:08 am
Solution:
365^365 ends in 5.

4 raised to odd power ends in 4 and 4 raised to even power ends in 6.
So 364^364 will end in 4^364 which in turn will end in 6.

363^363 ends in 3^363.

3^(4k+1) ends in 3 where k is 0,1, 2, 3…..
3^(4k+2) ends in 9.
3^(4k+3) ends in 7.
3^(4k+4) ends in 1.

363 = 4*90 + 3.

So 3^363 ends in 7.
Or 363^363 ends in 7.

362^362 ends in 2^362.

2^(4k+1) ends in 2 where k is 0, 1, 2, 3,….
2^(4k+2) ends in 4.
2^(4k+3) ends in 8.
2^(4k+4) ends in 6.

362 = 4*90 + 2.
Or 2^362 ends in 4
Or 362^362 ends in 4.

361^361 ends in 1^361 which ends in 1.
So 365^365+364^364+363^363+362^362+361^361 ends in last digit of (5 + 6 + 7 + 4 + 1 = 23) which is 3.

The units digit of 365^365+364^364+363^363+362^362+361^361 is 3.
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