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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.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [1]
In this case, s is called the least absolute remainder. [3] As with the quotient and remainder, k and s are uniquely determined, except in the case where d = 2n and s = ± n. For this exception, we have: a = k⋅d + n = (k + 1)d − n. A unique remainder can be obtained in this case by some convention—such as always taking the positive value ...
17 is divided into 3 groups of 5, with 2 as leftover. Here, the dividend is 17, the divisor is 3, the quotient is 5, and the remainder is 2 (which is strictly smaller than the divisor 3), or more symbolically, 17 = (3 × 5) + 2.
Three multiples can be subtracted (q 1 = 3), leaving a remainder of 21: 462 = 3 × 147 + 21. Then multiples of 21 are subtracted from 147 until the remainder is less than 21. Seven multiples can be subtracted (q 2 = 7), leaving no remainder: 147 = 7 × 21 + 0. Since the last remainder is zero, the algorithm ends with 21 as the greatest common ...
The final algorithm takes the six most significant bits of the size of the array, adds one if any of the remaining bits are set, and uses that result as the minrun. This algorithm works for all arrays, including those smaller than 64; for arrays of size 63 or less, this sets minrun equal to the array size and Timsort reduces to an insertion sort.
Introduced in Python 2.2 as an optional feature and finalized in version 2.3, generators are Python's mechanism for lazy evaluation of a function that would otherwise return a space-prohibitive or computationally intensive list. This is an example to lazily generate the prime numbers:
In a dynamically typed language, where type can only be determined at runtime, many type errors can only be detected at runtime. For example, the Python code a + b is syntactically valid at the phrase level, but the correctness of the types of a and b can only be determined at runtime, as variables do not have types in Python, only values do.