If xyz represents a three-digit number then xyzxyz is always divisible by:Options11 only7, 11 and 137 only11 and 13 only

Break Down the Problem

1. We need to analyze the number $xyzxyz$ which can be represented mathematically.
2. Determine the divisibility of $xyzxyz$ by the given options.

Relevant Concepts

1. $xyz$ is a three-digit number which can be expressed as $100x + 10y + z$.
2. Thus, $xyzxyz$ can be expressed as: $xyzxyz = 1000xyz + xyz = 1001 \times xyz$

Analysis and Detail

1. Since $xyzxyz = 1001 \times xyz$, we need to find the prime factorization of $1001$: $1001 = 7 \times 11 \times 13$
2. Therefore, $xyzxyz$ is divisible by $7$, $11$, and $13$.

Verify and Summarize

1. We have confirmed that $xyzxyz$, which is $1001 \times xyz$, is divisible by $7$, $11$, and $13$ since $1001$ has these numbers as its factors.
2. Among the options provided, the one that states $7, 11, \text{ and } 137$ is incorrect because $137$ is not a factor of $1001$.

Final Answer

The correct answer is that $xyzxyz$ is always divisible by 7, 11, and 13 only.