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What is the last digit of 207 4 35849 2074^{35849} ?

7 5 4 6

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3 solutions

Method 1:

207 4 35849 ( 2070 + 4 ) 35849 4 35849 (mod 10) 2074^{35849} \equiv (2070+4)^{35849} \equiv 4^{35849} \text{ (mod 10)} .

Now, we note that 4 n { 4 (mod 10) if n is odd. 6 (mod 10) if n is even. 4^n \equiv \begin{cases} 4 \text{ (mod 10)} & \text{if }n \text{ is odd.} \\ 6 \text{ (mod 10)} & \text{if }n \text{ is even.} \end{cases} .

Therefore, 207 4 35849 4 (mod 10) 2074^{35849} \equiv \boxed{4} \text{ (mod 10)} .


Method 2:

We need to find 207 4 35849 mod 10 2074^{35849} \text{ mod }10 . Since 2074 and 10 are not coprime integers, we have to consider 207 4 35849 mod 2 2074^{35849} \text{ mod }2 and 207 4 35849 mod 5 2074^{35849} \text{ mod }5 separately.

207 4 35849 0 (mod 2) 207 4 35849 207 4 35849 mod ϕ ( 5 ) (mod 5) Since gcd ( 2074 , 5 ) = 1 , Euler’s theorem applies. 207 4 35849 mod 4 (mod 5) Euler totient function ϕ ( 5 ) = 4 207 4 1 4 (mod 5) 5 n + 4 0 (mod 2) where n is an integer. n 0 207 4 35849 5 ( 0 ) + 4 4 (mod 10) \begin{aligned} 2074^{35849} & \equiv 0 \text{ (mod 2)} \\ 2074^{35849} & \equiv 2074^{\color{#3D99F6}35849 \text{ mod }\phi(5)} \text{ (mod 5)} & \small \color{#3D99F6} \text{Since } \gcd (2074, 5) = 1 \text{, Euler's theorem applies.} \\ & \equiv 2074^{\color{#3D99F6}35849 \text{ mod }4} \text{ (mod 5)} & \small \color{#3D99F6} \text{Euler totient function }\phi (5) = 4 \\ & \equiv 2074^1 \equiv 4 \text{ (mod 5)} \\ \implies 5 {\color{#3D99F6}n} +4 & \equiv 0 \text{ (mod 2)} & \small \color{#3D99F6} \text{where }n \text{ is an integer.} \\ \implies n & \equiv 0 \\ \implies 2074^{35849} & \equiv 5(0) + 4 \equiv \boxed{4} \text{ (mod 10)} \end{aligned}

We know that,

4 1 = 4 4^1 = 4

4 2 = 16 4^2 = 16

4 3 = 64 4^3 = 64

4 4 = 256 4^4 = 256

Observed that the last digit repeats in every cycle of 2 2 , that means that we must divide the exponent by 2 2 . If there is no remainder, the last digit is 6 6 . If the remainder is 1 1 the last digit is 4 4 .

Dividing 35849 35849 by 2 2 gives a remainder of 1 1 , so the last digit is 4 \boxed{4} .

Md Zuhair
Feb 8, 2017

2074 = 4 mod 10

Now 207 4 2 2074 ^2 = 6 mod 10

Again 207 4 3 2074 ^3 = 4 mod 10

Hence for odd n last digit is 4 \boxed{4}

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