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The main additions compared to AppleSoft BASIC on the Apple II were 19-digit long integer values in addition to floating point, better control over formatting input and output, and floppy disk commands that allowed file management without having to exit to Apple DOS. It included a rudimentary system to load and save fixed-width records, which ...
explicitly showing its relationship with Euclidean division. However, the b here need not be the remainder in the division of a by m. Rather, a ≡ b (mod m) asserts that a and b have the same remainder when divided by m. That is, a = p m + r, b = q m + r, where 0 ≤ r < m is the common 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]
As an example, VBA code written in Microsoft Access can establish references to the Excel, Word and Outlook libraries; this allows creating an application that – for instance – runs a query in Access, exports the results to Excel and analyzes them, and then formats the output as tables in a Word document or sends them as an Outlook email.
Conversely, Applesoft lacks the MOD (remainder) operator from Integer BASIC. Adapting BASIC for the Apple II was a tedious job as Apple received a source listing for Microsoft 6502 BASIC which proved to be buggy and also required the addition of Integer BASIC commands.
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.
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 ...
The remainder, as defined above, is called the least positive remainder or simply the remainder. [2] The integer a is either a multiple of d, or lies in the interval between consecutive multiples of d, namely, q⋅d and (q + 1)d (for positive q). In some occasions, it is convenient to carry out the division so that a is as close to an integral ...