%hw11.tex: Homework assignment 11
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\centerline
{
\bf Dr. Z.'s Number Theory Homework assignment 11
}

{\bf 1.} Using the first way,
find the unique $x$ between $0$ and $34$ such that

{\bf a.}
$$
x \equiv 4 \pmod {5} \quad , \quad
x \equiv 2 \pmod {7} \quad .
$$

{\bf b.}
$$
x \equiv 1 \pmod {5} \quad , \quad
x \equiv 6 \pmod {7} \quad .
$$

{\bf c.}
$$
x \equiv 2 \pmod {5} \quad , \quad
x \equiv 5 \pmod {7} \quad .
$$


{\bf 2.} Using the second way (the formula)
find the unique $x$ between $0$ and $34$ such that

{\bf a.}
$$
x \equiv 4 \pmod {5} \quad , \quad
x \equiv 2 \pmod {7} \quad .
$$

{\bf b.}
$$
x \equiv 1 \pmod {5} \quad , \quad
x \equiv 6 \pmod {7} \quad .
$$

{\bf c.}
$$
x \equiv 2 \pmod {5} \quad , \quad
x \equiv 5 \pmod {7} \quad .
$$

{\bf 3.} Using the make-a-table way,
find the unique $x$ between $0$ and $59$ such that

{\bf a.}
$$
x \equiv 3 \pmod {4} \quad , \quad
x \equiv 2 \pmod {3} \quad , \quad
x \equiv 3 \pmod {5} \quad .
$$

{\bf b.}
$$
x \equiv 2 \pmod {4} \quad , \quad
x \equiv 1 \pmod {3} \quad , \quad
x \equiv 4 \pmod {5} \quad .
$$

{\bf c.}
$$
x \equiv 3 \pmod {4} \quad , \quad
x \equiv 0 \pmod {3} \quad , \quad
x \equiv 1 \pmod {5} \quad .
$$


{\bf 4.} Using the formula,
find the unique $x$ between $0$ and $2001$ such that
$$
x \equiv 1 \pmod {2} \quad , \quad
x \equiv 5 \pmod {7} \quad , \quad
x \equiv 6 \pmod {11} \quad , \quad
x \equiv 4 \pmod {13} \quad .
$$


{\bf 5.} Use any method to find the
smallest non-negative integer $x$ such that
$$
x \equiv 3 \pmod {1024} \quad , \quad
x \equiv 3 \pmod {121} \quad , \quad
x \equiv 3 \pmod {169} \quad , \quad
x \equiv 3 \pmod {17} \quad , \quad
x \equiv 3 \pmod {529} \quad .
$$
\end
