Arithmetic: Properties Of Integers And Numbers Numbers And Digits The

Arithmetic: Properties of Integers and Numbers Numbers and Digits The decimal system is a system of tens, which uses ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). Each digit has a place value, depending on its position in the number. For example, 5968 is a four-digit number, with 5 being the first or thousands digit, 9 being the second or hundreds digit, 6 being the third or tens digit, and 8 being the fourth (last) or units digit. So, 5968 = (5 × 1000) + (9 × 100) + (6 × 10) + (8 × 1). Example Find a three-digit number whose first digit is 1. The second digit is twice the last, and the difference between the first two digits is 7. Solution The first digit is given as 1. Since the difference between the first two digits is 7, the second digit must be 8. Since the second digit is twice the last, the last digit is 4. Thus, the required three-digit number is 184. Integers, Natural Numbers and Whole Numbers The numbers {... −4, −3, −2, −1, 0, 1, 2, 3, 4, ...} form the set of integers. The notation { } indicates 'set' or collection, and the three dots indicate that the list continues endlessly to infinity. The numbers {1, 2, 3, 4, ...} form the set of positive integers (also called natural numbers or counting numbers). The numbers {−1, −2, −3, −4, ...} form the set of negative integers. The number 0 is an integer which is neither positive nor negative. It is usually not considered a counting number. The numbers {0, 1, 2, 3, 4, ...} form the set of whole numbers. Properties of the Integers 0 and 1 If 0 is added to or subtracted from any number, the number remains unchanged. Zero is therefore called the identity element for addition and subtraction . If 0 is multiplied by any number, the product is 0. If 0 is divided by any number (other than 0), the quotient is 0. Division by 0 is not defined because it has no meaning. If any number is multiplied or divided by 1, the number remains unchanged. One is therefore called the identity element for multiplication and division. Basic Operations The four basic operations

are addition (+), subtraction (−), multiplicaton (×) and division (÷). All four operations are binary operations, i.e., operations are performed with two numbers at a time to get a unique result. The result of addition is called the sum, the result of subtraction is called the difference, the result of multiplication is called the product, and the result of division is called the quotient. Subtraction is the inverse of addition, and division is the inverse of multiplication. The operations of addition, subtraction and multiplication on integers are said to be closed because the result of each operation is also an integer. For example, 5 + 8 = 13, 5 − 8 = −3, and 5 × 8 = 40. The operation of division on integers is not closed because the result is not necessarily an integer. For example, 5 ÷ 8 = 5/8 = 0.625 (which is not an integer). Quotients and Remainders Consider the following example. When 34 is divided by 6, the quotient is 5 and the remainder is 4 because 34 = 6 × 5 + 4. In general, when integer m (dividend) is divided by a non-zero integer n (divisor), then there exist a unique integer q (quotient) and a unique integer r (remainder) such that m = n × q + r. Note that m is divisible by n if and only if the remainder r is zero, i.e., m = n × q where q is an integer. For example, 36 is divisible by 6 because the remainder is 0 when 36 is divided by 6. Even and Odd Integers Even integers are those integers that are divisible by 2. The integers {... −4, −2, 0, 2, 4, ...} form the set of even integers. Odd integers are those integers that are not divisible by 2. The integers {... −5, −3, −1, 1, 3, 5, ...} form the set of odd integers. Note that the sum and difference of two even integers is even. The sum and difference of two odd integers is also even. The sum and difference of an odd integer and an even integer is odd. The product of two even integers is even. The product of two odd integers is odd. The product of an odd integer and an even integer is even. Consecutive Integers Consecutive integers are two or more integers in sequence, each of which is one more than the integer that precedes it. Consecutive integers may be represented by n, n + 1, n + 2, ..., where n is

an integer. For example, −3, −2, −1, 0, 1, 2, 3, 4 are consecutive integers. Consecutive even integers may be represented by 2n, 2n + 2, 2n + 4, ..., where n is an integer. For example, 0, 2, 4, 6, 8, 10, 12 are consecutive even integers. Consecutive odd integers may be represented by 2n + 1, 2n + 3, 2n + 5, ..., where n is an integer. For example, 5, 7, 9, 11, 13 are consecutive odd integers. Example The sum of three consecutive integers is less than 87. What is the maximum possible value of the smallest of the three integers? Solution Let the consecutive integers be n, n + 1, and n + 2. Then, their sum is n + (n + 1) + (n + 2) = 3n + 3. So, 3n + 3 < 87 ⇒ 3n < 84 ⇒ n < 28. Thus, the maximum possible value of n is 27. An alternative method to avoid the algebra and inequalities is given below. One-third of 87 is 29. So, let's guess the three consecutive integers to be 28, 29, 30. Their sum is 87. The sum must be less than 87. Therefore, the integers must be 27, 28, 29. Factors and Multiples If integer n divides integer m exactly with zero remainder, then n is a factor (divisor) of m, and m is divisible by n. For example, 16 is divisible by 2 but not by 3. The factors of 16 are 1, 2, 4, 8, and 16. If n is a factor of m, then m is called a multiple of n. Also, there exists another integer k such that m = n × k and thus m is also a multiple of k. For example, 16 = 2 × 8 and so 16 is a multiple of 2 as well as 8. Note that 8, 16, 24, 32, 40, 48, ... are multiples of 8; but 8, 24 and 40 are not multiples of 16. MUST-KNOW : Any integer is a multiple of each of its factors. Prime Numbers and Composite Numbers Any positive integer can be expressed as the product of 1 and itself. For example, 17 = 1 × 17; so, 1 and 17 are factors (divisors) of 17. A prime number is a positive integer that has exactly two different positive factors (1 and itself).

Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97 are the prime numbers upto 100. A composite number is a positive integer that has more than two different positive factors. Except 1, any number which is not a prime number is a composite number. The number 1 is neither prime nor composite because it has only one positive factor. MUST-KNOW: Any composite number can be uniquely expressed as a product of prime factors. To write a number as a product of primes, first divide the number by 2 as many times as possible, next divide the result by 3 as many times as possible, and then continue this procedure of dividing by the other successive prime numbers 5, 7, 11, 13 and so on, until all the factors are primes. For example to express 84 as a product of prime numbers, note that 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7.

Least Common Multiple or LCM A number k is a common factor of two numbers m and n if it is a factor of each of the numbers. For example, 5 is a common factor of 10 and 15 because 5 × 2 = 10 and 5 × 3 = 15. A number m is a common multiple of two numbers n and k if it is a multiple of each of the numbers. For example, 18 is a common multiple of 6 and 9 because 6 × 3 = 18 and 9 × 2 = 18. The smallest number that is a common multiple of two (or more) numbers is called the least common multiple (LCM). To find the LCM of given numbers, write each given number as a product of primes. Next, delete the common factors (if any) from all but one of the products. Finally, multiply the remaining factors to obtain the LCM. For example to find the LCM of 18, 45 and 81, each number is expressed as 18 = 2 × 3 × 3, 45 = 3 × 3 × 5 and 81 = 3 × 3 × 3 × 3. Next, 3 × 3 = 9 is the common factor; so, delete it in all but one of the products. Finally, the LCM is 2 × 3 × 3 × 3 × 3 × 5 = 810. Note that the LCM is the product of the highest power (i.e., number of times a factor occurs) of each separate factor. Example Professor

Morton grades a test paper

in 14 minutes, whereas Professor New stein takes only 12 minutes. If both professors start grading at 10:00 a.m., when will it be the first time they will finish grading a test paper at the same time? Solution Professor Morton will grade m papers in 14m minutes. Professor New stein will grade n papers in 12n minutes. To finish grading at the same time, 14m = 12n. Note that m and n must be integers because they denote number of test papers. Thus, the task is to find a common multiple of 14 and 12. Since the first time they finish together is needed, the LCM is required. Now, 14 = 2 × 7 and 12 = 2 × 2 × 3. When the common factor 2 is deleted in one of the products, the LCM is 2 × 2 × 3 × 7 = 84. The two professors will finish grading a test paper at the same time 84 minutes after starting, i.e., at 11:24 a.m.