Respuesta :
We start with
[tex]4^1 \equiv 4 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides
[tex]4^1 \equiv 4 \ (\text{mod} \ 10)[/tex]
[tex](4^1)^2 \equiv 4^2 \ (\text{mod} \ 10)[/tex]
[tex]4^2 \equiv 16 \ (\text{mod} \ 10)[/tex]
[tex]4^2 \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides again
[tex]4^2 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex](4^2)^2 \equiv 6^2 \ (\text{mod} \ 10)[/tex]
[tex]4^4 \equiv 36 \ (\text{mod} \ 10)[/tex]
[tex]4^4 \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides again
[tex]4^4 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex](4^4)^2 \equiv 6^2 \ (\text{mod} \ 10)[/tex]
[tex]4^8 \equiv 36 \ (\text{mod} \ 10)[/tex]
[tex]4^8 \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides again
[tex]4^8 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex](4^8)^2 \equiv 6^2 \ (\text{mod} \ 10)[/tex]
[tex]4^{16} \equiv 36 \ (\text{mod} \ 10)[/tex]
[tex]4^{16} \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
So far we have found that,
[tex]4^1 \equiv 4 \ (\text{mod} \ 10)[/tex]
[tex]4^2 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex]4^4 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex]4^8 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex]4^{16} \equiv 6 \ (\text{mod} \ 10)[/tex]
Multiply out all of the left sides of those equations above. Multiply out the right sides as well to balance things out
[tex](4^1)*(4^2)*(4^4)*(4^8)*(4^{16}) \equiv 4*6*6*6*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{1+2+4+8+16} \equiv 24*36*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 4*6*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 24*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 4*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 24 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 4 \ (\text{mod} \ 10)[/tex]
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The final answer is 4
[tex]4^1 \equiv 4 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides
[tex]4^1 \equiv 4 \ (\text{mod} \ 10)[/tex]
[tex](4^1)^2 \equiv 4^2 \ (\text{mod} \ 10)[/tex]
[tex]4^2 \equiv 16 \ (\text{mod} \ 10)[/tex]
[tex]4^2 \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides again
[tex]4^2 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex](4^2)^2 \equiv 6^2 \ (\text{mod} \ 10)[/tex]
[tex]4^4 \equiv 36 \ (\text{mod} \ 10)[/tex]
[tex]4^4 \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides again
[tex]4^4 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex](4^4)^2 \equiv 6^2 \ (\text{mod} \ 10)[/tex]
[tex]4^8 \equiv 36 \ (\text{mod} \ 10)[/tex]
[tex]4^8 \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
Square both sides again
[tex]4^8 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex](4^8)^2 \equiv 6^2 \ (\text{mod} \ 10)[/tex]
[tex]4^{16} \equiv 36 \ (\text{mod} \ 10)[/tex]
[tex]4^{16} \equiv 6 \ (\text{mod} \ 10)[/tex]
----------------------------
So far we have found that,
[tex]4^1 \equiv 4 \ (\text{mod} \ 10)[/tex]
[tex]4^2 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex]4^4 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex]4^8 \equiv 6 \ (\text{mod} \ 10)[/tex]
[tex]4^{16} \equiv 6 \ (\text{mod} \ 10)[/tex]
Multiply out all of the left sides of those equations above. Multiply out the right sides as well to balance things out
[tex](4^1)*(4^2)*(4^4)*(4^8)*(4^{16}) \equiv 4*6*6*6*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{1+2+4+8+16} \equiv 24*36*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 4*6*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 24*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 4*6 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 24 \ (\text{mod} \ 10)[/tex]
[tex]4^{31} \equiv 4 \ (\text{mod} \ 10)[/tex]
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The final answer is 4
