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Problem 35 (Connecting Multiples And Spans Of Integers)

Suppose that $a$ and $b$ are integers.

  1. Prove that if $a$ is a multiple of $b$, then we must have $\text{span}(a)\subseteq\text{span}(b)$.
  2. Is the converse of the statement above true or false? Make sure you prove your result.

Solution

1. Suppose $a,b\in\mathbb{Z}$ and let $a$ be a multiple of $b$. This means $a=sb$ for some $s\in\mathbb{Z}$. We now prove that $\text{\span}(a)\subseteq \span(b)$. First we must show that $\span(a)\subseteq \span(b)$. Let $p\in \span(a)$, which means $p=ja$ where $j\in\mathbb{Z}$. Recall $a=sb$ such that $p=jsb$. This means $p\in \span(b)$. Thus, $\span(a)\subseteq \span(b)$.

2. Supposed $a,b\in\mathbb{Z}$ and let $\span(a)\subseteq \span(b)$. We'll show that $a$ is a multiple of $b$. First, let $a\in \span(a)$. Since $\span(a)\subseteq \span(b)$, then it follows that $a\in \span(b)$. This means $a=kb$ for some $k\in\mathbb{Z}$. Thus, by the definition of multiple, $a$ is a multiple of $b$.

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