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$\{1\} \subset \mathbb{Z}/m\mathbb{Z}$ is spanning, but not independent. However, you can still use $1$ in the way you want to. You don't need to use linear independence; all that you need is that it generates the module. $\endgroup$ –
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To expand an expression using the distributive property, multiply each term inside a set of parentheses by each term outside the parentheses, and then simplify by combining like terms. The FOIL method is a mnemonic used to help remember the steps for multiplying two binomials.
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