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While original letters written by Nehru were in English, they were translated into Hindi by the Hindi novelist Munshi Premchand under the name Pita Ke Patra Putri Ke Naam. [citation needed] In 2014, a Spanish translation with the title "Cartas a mi hija Indira" (Letters to my daughter Indira), was released by Rodolfo Zamora. Five additional ...
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
Historically the term was used for a number that was a multiple of 1, [10] [11] or to the whole part of a mixed number. [12] [13] Only positive integers were considered, making the term synonymous with the natural numbers. The definition of integer expanded over time to include negative numbers as their usefulness was recognized. [14]
More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function (). This can be used to solve change-making problems (where the set T specifies the available coins).
The equivalence class modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted as (a mod m), or as a or [a] when the modulus m is known from the context.
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Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8.
Conversely, given a solution to the SubsetSumZero instance, it must contain the −T (since all integers in S are positive), so to get a sum of zero, it must also contain a subset of S with a sum of +T, which is a solution of the SubsetSumPositive instance. The input integers are positive, and T = sum(S)/2.