Congruences

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Name:  Marshal Thrasher

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Congruence 1.  Find an integer $x$ such that $4^{128} \equiv x$ $mod 9$ and $0 \leq x \leq 100$.

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\hspace{.5in} If we begin with $4 \equiv 4$ $mod9$ and also, $4^2 \equiv 7$ $mod9$, then by Proposition 21, we can say:
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$4^3 \equiv 28$ $mod9$ (where $1$ also works for $28$)
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If we repeat the original process with $4^3$ until we get to $4^{126}$, we can then utilize the $4$ and $4^2$ values respectively:

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$4^{127} \equiv 4$ $mod9$ 
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and finally 
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$4^{128} \equiv 7$ $mod9$.
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Congruence 2. Find an integer $y$ such that $3^{128} \equiv y$ $mod 4$ and $0 \leq y \leq 3$.
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\hspace{.5in} If we begin with \\
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$3 \equiv 3$ $mod4$, \\ 
$3^2 \equiv 1$ $mod4$, \\ 
$3^3 \equiv 3$ $mod4$, \\ 
$3^4 \equiv 1$ $mod4$, \\ 
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   then by Proposition 21, we can say:
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$3^6 \equiv 1$ $mod4$, \\ 
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	Considering $128$ is an even integer and $3^2 \equiv 1$ $mod4$, we can do this until we get to $3^{128} \equiv 1$ $mod4$.

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Congruence 3. For each of the following congruence's, find integers $x_{i}$ such that $0 \leq x_{i} \leq 6$ that satisfy the congruence.
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