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The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of ...
If that number is an even natural number, the original number is divisible by 4 Also, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4. Example. General rule
In particular, for a prime number p we have the explicit formula r 4 (p) = 8(p + 1). [1] Some values of r 4 (n) occur infinitely often as r 4 (n) = r 4 (2 m n) whenever n is even. The values of r 4 (n) can be arbitrarily large: indeed, r 4 (n) is infinitely often larger than . [1]
a perfect number equals the sum of its proper ... a highly abundant number has a sum of positive divisors that is greater ... 4, 5, 8, 10, 20, 25, 40, 50, 100, 200 12 ...
Columnar addition – the numbers in the column are to be added, with the sum written below the underlined number. There are also situations where addition is "understood", even though no symbol appears: A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. [4]
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base 6, 8, and 12. Every centered square number except 1 is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean ...
Coverage of this topic in Smithsonian magazine describes the Numberphile video as misleading and notes that the interpretation of the sum as − + 1 / 12 relies on a specialized meaning for the equals sign, from the techniques of analytic continuation, in which equals means is associated with. [35]