For each of the following partial inventories, copy data on a suitable blank, extend, and find the total values. PROBLEMS 1. The table below gives the total number of freight trains and the total amount of freight hauled on a certain road in the years indicated. Find the average number of tons of freight per train for each of the years named. Also find the average load per train for this whole period. Suggestion. To find the average load per train for the period, find first the total number of trains and the total amount of freight carried. 2. The attendance at a certain school for the twenty school days of a month was: 1874, 1892, 1924, 1914, 1903, 1917, 1893, 1897, 1898, 1898, 1902, 1904, 1907, 1899, 1897, 1904, 1914, 1891, 1907, 1894. Find the average attendance for this month. 3. The following table gives the number of acres in corn and the total yield for the states named as given by a recent census. Find to the nearest bushel the average yield in each state. Also find the average for all these states. Suggestion. To find the average for all these states find first the total number of acres and the total yield. CHAPTER VII DIVISIBILITY OF NUMBERS, FACTORS, MULTIPLES This chapter serves as an introduction to the chapters on fractions, since dealing with fractions requires a knowledge of the divisibility of numbers, and of the factors and multiples of numbers. 92. Integers. The numbers obtained by ordinary counting, beginning with 1, are called integers. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are all the integers from 1 to 10 inclusive. 93. Odd and Even Integers. Every second integer, beginning with 2, is called an even integer. Zero is also regarded as an even integer. Thus, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 are all the even integers from 0 to 20 inclusive. All integers which are not even are called odd integers. We frequently say odd or even numbers, instead of odd and even integers. 94. Factors. If a given integer is the product of two or more integers, these are said to be factors of the given integer. Thus, 2 and 3 are factors of 6, as are also 1 and 6. All 95. Prime and Composite Numbers. A number (integer) which has no factors except itself and 1 is called a prime number. integers which are not prime are called composite numbers. Thus, 1, 2, 3, 5, 7, 11 are prime numbers, while 4, 6, 8, 9, 10 are composite. 96. Prime Factors. A prime number which is a factor of another number is called a prime factor of that number. Thus, 1, 2, 3 are prime factors of 6, and also of 12. ORAL EXERCISES 1. Give all prime numbers from 1 to 50. 2. Give all prime factors of the numbers: 6, 8, 10, 15, 18, 20. 3. Give all composite factors of 12, 16, 24, 36. 97. Divisibility. - A number is said to be divisible by any one of its factors. 98. Tests of Divisibility. - A number is divisible by 2 if its last digit is divisible by 2. Zero is here regarded as divisible by 2. Thus 364, 748, 196, 48780 are divisible by 2. A number is divisible by 4 if the last two of its digits represent a number which is divisible by 4. Thus 324, 25632, 70976, 134568 are all divisible by 4. A number is divisible by 8 if the last three of its digits represent a number which is divisible by 8. Thus, 7144, 245632, 19248, 3749672 are all divisible by 8. All numbers which end in 5 or 0 are divisible by 5. All numbers which end in 0 are divisible by 10. A number is divisible by 3 if the sum of its digits is divisible by 3. Thus, 432423 is divisible by 3 because 4 + 3 + 2 + 4 + 2 + 3 divisible by 3. = 18 is A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 6 if it is even and is divisible by 3. Thus, 48732 is divisible by 6, while neither 3442 nor 753 is divisible by 6. A number is divisible by 12 if it is divisible by both 3 and 4. 99. Divisors. If a given number is divisible by a number, then this second number is called a divisor of the given number. ORAL EXERCISES State which of the numbers 1, 2, 3, 4, 5, 6, 8, 10, 12, if any, are divisors of each of the following numbers: 100. Factoring. — Factoring a number consists in finding all the prime factors of the number. The product of all the prime factors of a number is the number itself. Thus, the prime factors of 5 are 1 and 5, and 1 × 5 factors of 6 are 1, 2, and 3, and 1 × 2 × 3 = 6. = 5. The prime Sometimes one number is used as a factor several times. Thus, 2 is used as a factor 3 times in 8, because 2 × 2 × 2 = 8. 101. Common Factors. If a number is a factor of each of two or more numbers, it is said to be a common factor of these numbers. Thus, 2 is a common factor of 6, 8, and 10. 3 is a common factor of 9, 18, 21; and 7 is a common factor of 28 and 42. 102. Greatest Common Divisor. The greatest number which is a common factor of two or more numbers is called the Greatest Common Divisor (G. C. D.) of these numbers. The G. C. D. of two or more numbers is the product of all the prime common factors of these numbers. Example. Find the G. C. D. of 32, 64, 96, 128. The work is conveniently arranged thus: Divide each number by the common factor 2, obtaining 16, 32, 48, 64 as the quotients. 2 is a common factor of the quotients. Continuing in this manner, we finally get 1, 2, 3, 4 as quotients, and these have no common factor except 1. Hence, the G. C. D. = 2 × 2 × 2 × 2 × 2 = 32. EXERCISES = 1. Find all prime factors of: 6, 8, 10, 12, 16, 18, 21, 24, 32, 36, 42. 2. Find all prime common factors of 18, 24, 36, 42. Find the G. C. D. of each of the following sets of numbers: 3. 30, 42, 64, 72 4. 16, 96, 108, 124 9. 17, 51, 85, 102 |