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  2. Euclid's lemma - Wikipedia

    en.wikipedia.org/wiki/Euclid's_lemma

    Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b. For example, if p = 19 , a = 133 , b = 143 , then ab = 133 × 143 = 19019 , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well.

  3. Euclidean division - Wikipedia

    en.wikipedia.org/wiki/Euclidean_division

    In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder ...

  4. Euclidean algorithm - Wikipedia

    en.wikipedia.org/wiki/Euclidean_algorithm

    Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. [118] For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common ...

  5. Fundamental theorem of arithmetic - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of...

    (In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as Euclid's lemma, and it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. —

  6. Euclid's theorem - Wikipedia

    en.wikipedia.org/wiki/Euclid's_theorem

    Filip Saidak gave the following proof by construction, which does not use reductio ad absurdum [15] or Euclid's lemma (that if a prime p divides ab then it must divide a or b). Since each natural number greater than 1 has at least one prime factor , and two successive numbers n and ( n + 1) have no factor in common, the product n ( n + 1) has ...

  7. Division algorithm - Wikipedia

    en.wikipedia.org/wiki/Division_algorithm

    Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.

  8. Greatest common divisor - Wikipedia

    en.wikipedia.org/wiki/Greatest_common_divisor

    The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.

  9. Euclidean - Wikipedia

    en.wikipedia.org/wiki/Euclidean

    Euclid's lemma: if a prime number divides a product of two numbers, then it divides at least one of those two numbers; Euclidean domain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the extended Euclidean algorithm to work