Integer Division

Each member of a group chooses one of these four lists of integers to investigate and answers the following questions. 

The group members now compare notes. What is the common thread in each of your answers to the questions from part (1)7 

The number 330 is in all four lists. What properties does it have that cause it to be in each list Express 330 as a product of smaller integers. 

As mentioned in the previous section, given two numbers (12 and 3, for example), they can both be added together (12 + 3 = 15), one can be subtracted from the other (12 - 3 = 9), one can be divided by the other (12 + 3 = 4), or they can be multiplied together (12 3 = 36). For the moment, division involving two integers will be explored. 

Another way to determine that 12/3 = 4 is by using long division. Clearly, 3 goes into 12 a total of 4 times with a remainder of 0  

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Obviously Eq. (1) holds only because the remainder is discarded in integer division performed by the computer. 

This chapter begins with an investigation of integer division and modulo arithmetic. We then explore check digit schemes that employ the number theoretic concepts we have developed. The chapter ends with an application of these concepts to cryptography. 

In terms of integer division, some special integers need to be mentioned. They are defined next. 

An integer divisible only by exacdy two integers, itself and 1, is called a prime number. The number 1 is conventionally excluded since it has only one factor. 

Note that the result of integer division (as in the last line above) is rounded down. To find the remainder from such a division, the modulus operator, 7o, is used for example, the result of the operation 14 % 4, is 2. The operators ++, and — can be used as a shorthand for increment and decrement respectively. For example, the statement a++ , is equivalent to the statement a = a + 1 . Further, there exists a set of compound assignment operators which take the form op=. As a demonstration of their usage, the following two statements are equivalent ... 

Note that i%2 and j%2 are integer divisions, returning only the value of the integer part of the result. For instance, 4%3 = 1. 

The alarm message for divisions by zero is division by zero [ interval ], where type can be either integer or float. A simple example of an integer division by zero alarm is shown below. 

Integer division by zero fnel rate ALARM (A) integer division by zero [0, 65535] at fuelratecontroller.c 9596.27-9597.66 unreachable(inf)... 

Furthermore, actual designs will normally observe the pinch division. Hence A shells should be evaluated and taken as the next largest integer for each side of the pinch. The number-of-shells target is then... 

Integers and exact numbers In multiplication or division by an integer or an exact number, the uncertainty of the result is determined by the measured value. Some unit conversion factors are defined exactly, even though they are not whole numbers. For example, 1 in. is defined as exactly 2.54 cm and the 273.15 in the conversion between Celsius and Kelvin temperatures is exact so 100.000°C converts into 373.150 K. 

Take a moment to look at the flowcharts. As you can see at a glance, the multiplication and division rules are much easier to remember than addition and subtraction. It is a good idea to be proficient in integer arithmetic, in both speed and accuracy. The best way is to practice. It is just like learning to ride a bicycle. At first it seems so difficult, and then with practice you are riding without even thinking. As you are starting out with your review, follow the flowchart with each problem. Soon the flowchart will become second nature to you. 

As in the fifth division, must be a positive integer and cannot sO, whilst may be either hydrogen or a monad alcohol... 

Division is usually the last of the four basic operations that kids study in school. Why Because many of the results are not whole numbers. When you add, subtract, or multiply whole numbers together, you always get a whole number as a result (or an integer — in the case of subtracting a larger number from a smaller number). Not so with division. Not every division problem comes out evenly, and dealing with a remainder can be a bit unsettling or even daunting. 

Numbers may be combined using the arithmetic operations of addition ( + ), subtraction (—), multiplication (x) and division (/ or -). The type of number (integer, rational, irrational) is not necessarily maintained under combination. Thus, for example, addition of the fractions 1/4 and 3/4 yields an integer, but division of 3 by 4 (both integers) yields the rational number (fraction) 3/4. When a number (say, 8) is multiplied by a fraction (say, 3/4), we say in words that we want the number which is three quarters of 8 which, in this case, is 6. 

A The value off(x) is indeterminate for x = 4 and x = — 3 because division by zero yields an indeterminate result. For all other values of x the function is defined, and consequently the domain of ffx) could be either all real numbers, excluding x — 4 and x = —3, or all integers excluding x = 4 and x= —3. We could write this explicitly as either ... 

If there is no remainder, the integer is said to be divided evenly, or divisible by the number. 

Jig. 2. A sine-wave carrier (frequency ojq) is assumed to be periodically amplitude modulated at frequency 2 with a Gaussian envelope. Exact frequency division (ioq/ 2 = integer) is obtained if the phase of the carrier is fixed to the phase of the modulation envelope. 

Theorem 1. Let C be a cycle of a benzenoid system. The size of C is necessarily an even integer. If the size of C is divisible by four, then in the interior of C there is an odd number of vertices. Otherwise, in the interior of C there are either no vertices or their number is even. 

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