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# Category Archives: Theorems

## If *a* divides *b* and *a* divides *c*, then *a* divides *(b-c)*

In reading about Euclid’s proof of the infinitude of prime numbers, the only part that wasn’t completely clear to me was this: If \(p\) divides \(P\) and \(q\), then \(p\) would have to divide the difference of the two numbers, … Continue reading

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## Proof that if *f* and *g* are continuous functions, then *f/g* is also continuous (as long as *g(x)* ≠ 0)

In almost any calculus or analysis textbook, in the chapter on continuity of functions, you’ll encounter four theorems about the operations on functions that preserve continuity: multiplying a continuous function by a scalar (real number), adding two continuous functions, multiplying … Continue reading

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## The infinitude of prime numbers—Euclid’s proof in my own words

Euclid is believed to be the first mathematician to prove that there are infinitely many prime numbers. Most of us learn only that Euclid established and codified the framework of two- and three-dimensional geometry, but he accomplished far more than … Continue reading

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