Web Reference: Proof: We need to argue two things. First, we need to show that $q$ and $r$ exist. Then, we need to show that $q$ and $r$ are unique. To show that $q$ and $r$ exist, let us play around with a specific example first to get an idea of what might be involved, and then attempt to argue the general case. a = bq + r: If the integer c divides a and b, then by properties of division, it would divide also r = a bq. In other words, any integer that is a common divisor of two numbers a; b (b > 0), is also a divisor of the remainder of the division r of a by b. It was proved in an example of Module 3.2 Methods of Proof and exploited in another example of that module. If the number 253 is composite, for example, it must have a factor less than or equal to 15.
YouTube Excerpt: In this video, we try to understand and
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