To begin, find the remainder when 54 is divided by 17.
R(54/17) = 3. So our question can be reduced to what is the remainder when 3^124 is divided by 17.
Let us look for the cyclicity then.
R(3/17) = 3
R(9/17)=9
R(27/17)=10
R(81/17)=13
R(243/17) = 5
R(729/17)=15
R(2187/17)=11
R(6561/17)=16
R(19683/17)=14
R(59049/17)=8
R(177147/17)=7
R(531441/17)=4
R(1594323)=12
R(4782969/17)=2
R(14348907/17)=6
R(43046721/17)=1
R(387420489/17)=3

Finally, we have a cycle with cyclicity 16. So Remainder of 3^124 divided by 17 will be equal to 3^Remainder of 124/16 divided by 17.
R(124/16) = 12
So R(3^12)/17 =4
Hence Answer will be 4

Now a trick to solve this kind of question. The maximum cyclicity a unit digit has is 4 and the maximum cyclicity a remainder can have is 16. So instead of doing this calculation, we can safely use 16 as cyclicity and all the remainders with cyclicity of 4 and 8 will also have a cyclicity of 16.
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Asked: What is the remainder when \(54^{124}\) is divided by 17?

Remainder when 54^{124} is divided by 17
= remainder when 3^{124} is divided by 17
= remainder when (-4)^{31} is divided by 17
= remainder when (-1)^{15}×(-4) is divided by 17
= 4

IMO A

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What is the remainder when \(54^{124}\) is divided by 17?

\(54^{124}\) = \(3^{124}\) mod 17

\(54^{124}\) = \(9^{62}\) mod 17

\(54^{124}\) = \(81^{31}\) mod 17

\(54^{124}\) = \(-4^{31}\) mod 17

\(54^{124}\) = (\(-4^{1}\)) * (\(-4^{30}\)) mod 17

\(54^{124}\) = (\(-4^{1}\)) * (\(16^{15}\)) mod 17

\(54^{124}\) = (-4) * (\(-1^{15}\)) mod 17

\(54^{124}\) = (-4*-1) mod 17

\(54^{124}\) = 4 mod 17 = 4

Ans = A
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Given, (54^124)/17 or, [(51+3)^124]/17

So, it will have same remainder as [(3^4)^31]/17 or, [81^31]/17
or, [(-4)^31]/17 or, [(-4)*(-4)^30]/17 or, [(-4)*(16)^15]/17 (Trying to reach to -1 or +1)

It has same remainder as [(-4)*(-1)^15]/17 = (-4)*(-1) = 4.

So, I think A.

Very efficient method, thanks!

chetan2u what are the odds of such a question coming up, considering we have to calculate the cyclicity upto 16 levels, as per the first method?

I did not understand the method how are we going to -4 and its degrees, can someone explain the question and how it will be solved. VeritasKarishma

First check: https://www.veritasprep.com/blog/2011/0 ... ek-in-you/

\(54^{124} = (51 + 3)^{124}\)

Since 51 is divisible by 17, remainder when divided by 17 will be \(3^{124}\).

\(3^{124} = (3^4)^{31} = 81^{31} = (85 - 4)^{31}\)

Since 85 is divisible by 17, remainder will be \((-4)^{31}\).


\((-4)^{31} = -4*(4)^{30} = -4*(16)^{15} = -4*(17 - 1)^{15}\)

Remainder on division by 17 will be -4*-1 = 4

Answer (A)
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