Find the last 2 digits of 2 1 9 9 9 .
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You have a typo in your solution. It should be 2 1 9 9 9 m o d 2 0 instead of 2 1 9 9 9 m o d ϕ ( 2 0 ) .
hi i want to improve in these types of sums as i am very bad at these. can u suggest some tips or website to improve
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I learned Number Theory in Brilliant. You can select Number Theory (or other topics) under the Topics menu. There are learning notes (wikis) with examples and problems.
Can you explain 3rd and 4th step
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Euler's theorem: a ϕ ( n ) ≡ 1 ( m o d n ) , if a and n are coprime.
The number given : 2^{1999} This number can also be written as → (2^{10})^{199}^{9}. Point to be noted that 2^{10}^{a} is of the form you find two conditions arises if a is even then then the two units place will be 76 but if its odd then the two units place will be 24. Hence the solution becomes → 24 ×2^{9}=512×24=12288 hence answer is 88.... Q.E.D.
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We need to solve for 2 1 9 9 9 ( m o d 1 0 0 ) . Let us consider 2 1 9 9 9 ( m o d 4 ) and 2 1 9 9 9 ( m o d 2 5 ) separately.
2 1 9 9 9 ≡ 2 ⋅ 4 9 9 9 ≡ 0 ( m o d 4 )
2 1 9 9 9 ≡ 2 1 9 9 9 m o d ϕ ( 2 5 ) ( m o d 2 5 ) Since g cd ( 2 , 2 5 ) = 1 , we can use Euler’s theorem. ≡ 2 1 9 9 9 m o d 2 0 ( m o d 2 5 ) ≡ 2 1 9 ( m o d 2 5 ) ≡ 1 0 2 4 ⋅ 5 1 2 ( m o d 2 5 ) ≡ ( − 1 ) ⋅ ( 1 2 ) ( m o d 2 5 ) ≡ 8 8 ( m o d 2 5 )
Since 8 8 ≡ { 0 ( m o d 4 ) 8 8 ( m o d 2 5 ) ⟹ 2 1 9 9 9 ≡ 8 8 ( m o d 1 0 0 )