What are the last two digits of 2017^{2017}?

[nanase]

I need help in working this out

No other hint is given. What I tried is looking for pattern by typing it to the power of 1 , 2 and 3 and so on, but it gets too large and incomputable. What other way can I use and how? Thank you

 

[BigBeachBanana]

I need help in working this out
View attachment 36674
No other hint is given. What I tried is looking for pattern by typing it to the power of 1 , 2 and 3 and so on, but it gets too large and incomputable. What other way can I use and how? Thank you

Hint: Modulo 100

 

[nanase]

Hint: Modulo 100

Hi BBB
I tried this which gives 17, then I search for pattern
like 17^1 , 17^2 and so on, but the last two digits don't form any distinguishable pattern. Also I am not supposed to use calculator for this? can you give more hints/help?

 

[BigBeachBanana]

Hi BBB
I tried this which gives 17, then I search for pattern
like 17^1 , 17^2 and so on, but the last two digits don't form any distinguishable pattern. Also I am not supposed to use calculator for this? can you give more hints/help?

$$17^{17} = 17^{16}\times 17$$
Can you figure out the following even powers?
$17^{2},17^{4},17^{8},17^{16} \mod 100?$

Edit:
Use $17^2 \mod 100$ to figure out $17^4 \mod 100$ and so on...

 

[nanase]

The only thing I understand is the indices property applied there, but I am lost can you help further?

 

[BigBeachBanana]

The only thing I understand is the indices property applied there, but I am lost can you help further?

What is $17^2 \mod 100?$

 

[nanase]

What is $17^2 \mod 100?$

89

 

[BigBeachBanana]

89

$17^2 \equiv 89 \equiv -11 \mod 100$
$(17^2)^2 \equiv (-11)^2 \equiv 21 \mod 100$
Continue until $17^{16}$

 

[nanase]

I can't continue because the calculator won't display the last 2 digits anymore.
but I am not supposed to use calculator so this is probably not the right method also. But what am I seeing here? I don't see any pattern yet

 

[BigBeachBanana]

View attachment 36679
I can't continue because the calculator won't display the last 2 digits anymore.
but I am not supposed to use calculator so this is probably not the right method also. But what am I seeing here? I don't see any pattern yet

Read post#8 again. You're not looking for a pattern, but leveraging prior modulo to find the subsequent ones. We don't need all of the even powers, only 2,4,8,16. You shouldn't need a calculator, only multiply 2 digits by 2 digits.

 

[nanase]

I re-do the above working

please the me if this is correct, the last 2 digits are 77.
I use all the hints given above.

 

[BigBeachBanana]

I re-do the above working
View attachment 36680
please the me if this is correct, the last 2 digits are 77.
I use all the hints given above.

The final answer is correct but your intermediate steps are not.
$(17^4)^2 \neq 17^6$

 

[nanase]

The final answer is correct but your intermediate steps are not.
$(17^4)^2 \neq 17^6$

Thank you for correcting my steps, appreciate it as I am here not just for the answer, but for understanding how things work also mathematically
Thank you muchhhh Mr BBB.
Is the above correct ?

 

[Steven G]

I suspect that you will NOT find a pattern with the last two digits of powers of 17. mod 100 will help.

Why are you looking at all even powers of 17?

Only consider 17, 17^2, 17^4, 17^8, ....
How do you go from one to the next?

 

[Steven G]

View attachment 36690
Thank you for correcting my steps, appreciate it as I am here not just for the answer, but for understanding how things work also mathematically
Thank you muchhhh Mr BBB.
Is the above correct ?

Maybe I missing something but I don't think your answer is correct as I don't see your answer.
An answer should look like the last two digits in 2017^2017 is _ _ !!!!!!

 

[nanase]

Maybe I missing something but I don't think your answer is correct as I don't see your answer.
An answer should look like the last two digits in 2017^2017 is _ _ !!!!!!

in post #11 I put 77, Mr Banana is correcting me on my steps.

 

[Steven G]

in post #11 I put 77, Mr Banana is correcting me on my steps.

No, I won't accept your reply--sorry
In post 13 you asked if the work in that post is correct.
If someone gives me some work to look at I would NEVER say that it is correct when it is incorrect just because the work they showed me before was correct.

 

[nanase]

No, I won't accept your reply--sorry
In post 13 you asked if the work in that post is correct.
If someone gives me some work to look at I would NEVER say that it is correct when it is incorrect just because the work they showed me before was correct.

noted, you have a point. Thank you so much for the advise

 

[pka]

I need help in working this out
View attachment 36674
No other hint is given. What I tried is looking for pattern by typing it to the power of 1 , 2 and 3 and so on, but it gets too large and incomputable. What other way can I use and how? Thank you

D you read reply #2? SEE HERE

 

[nanase]

D you read reply #2? SEE HERE

yes I do , but i cant use calculator or technology