70. Common burning gas is .97 of the weight of air, and a liter of air weighs 1.2938. In a shop there are 65 jets, each one of which burns 123' an hour, and is used 5 hours in the winter evenings. Calculate the weight of the gas used in a month, and the expense of lighting the shop, when gas costs 6 cents a cubic meter. 71. A merchant buys one kind of wine at 30 cents a liter, another kind at 21 cents a liter; he mixes the two kinds by putting 5' of the first with 8' of the second. For how much a liter must he sell the mixture in order to gain $3.75 a hektoliter? 72. If it requires 360 tiles to drain an ar of land, what will it cost to drain 17.784ha, when the tiles cost $20 a thousand, and the expense of laying is the same as the cost of the tiles? 73. It is found in building that hewn stone of medium durability ought not to support, as a permanent weight, more than .07 of the weight that it would require to crush it. A certain kind of stone used for building will be crushed under a weight of 250ks a square centimeter. What is the greatest height to which a wall constructed of this material can be safely carried, the specific gravity of the stone being 2.1? 74. Several different kinds of wine are mixed as follows 245' at 20 cents a liter, 5471 at 15 cents a liter, 3441 at 25 cents a liter. How much does the mixture cost a liter? : 75. A farmer wishes to drain a field of 8.75ha. Each hektar requires 750m of ditches. The opening of these ditches costs 10 cents a running meter; the tiles are 30cm long and cost $15 a thousand. He pays 2 cents a meter for laying the tiles, and 4 cents a meter for filling the ditches. What is the cost of draining the field? CHAPTER X. MULTIPLES AND MEASURES OF NUMBERS. 216. If one number can be divided by another number, without remainder, the divisor is called a factor, or measure, of the dividend, and the dividend a multiple of the divisor. Thus, 35 can be divided by 5 without remainder; therefore 5 is called a factor or measure, of 35, and 35 a multiple of 5. 217. Numbers which can be divided, without remainder, only by themselves and 1, are called prime numbers. The smaller ones are easily found by trial, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc. 218. Other numbers are each the product of a fixed set of prime factors, and are called composite numbers. 219. Numbers divisible by 2 are called even numbers. All other numbers are odd numbers. All primes (except 2) are odd. 220. Write a series of natural numbers in order; cancel the even ones; then place a dot over the multiples of 3: you produce this result: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, etc. Each multiple of 6 has both the dot and the cancelling line, and the only numbers without the dot or line come just before or just after the multiples of 6. Therefore, If a prime number be divided by 6 the remainder must be either 1 or 5. Hence, to separate a composite number into its prime factors, Divide the given number by any prime number that is contained in it without remainder, then the quotient by any prime number that is contained in it without remainder, and so on until the quotient is itself a prime number. The several divisors and the last quotient are the prime factors required. 222. The following tests are very useful for determining without actual division whether a number contains certain factors: 1. A number is divisible by 2 if its last digit is even. 2. A number is divisible by 4 (22) if the number denoted by the last two digits is divisible by 4. 3. A number is divisible by 8 (23) if the number denoted by the last three digits is divisible by 8. 4. A number is divisible by 3 if the sum of its digits is divisible by 3. 5. A number is divisible by 9 (32) if the sum of its digits is divisible by 9. 6. A number is divisible by 5 if its last digit is either 5 or 0. 7. A number is divisible by 25 (52) if the number denoted by the last two digits is divisible by 25. 8. A number is divisible by 125 (53) if the number denoted by the last three digits is divisible by 125. 9. A number is divisible by 6 if its last digit is even and the sum of its digits is divisible by 3. 10. A number is divisible by 11 if the difference between the sum of the digits in the even places and the sum of the digits in the odd places is either 0 or a multiple of 11. NOTE. The shortest method of dividing by 25 is to multiply by 4 and divide by 100; by 125, is to multiply by 8 and divide by 1000. In adding the digits of a number to determine whether their sum is a multiple of a certain number, omit those digits which are seen at a glance to be multiples of the number. Thus, to discover whether 8,983,167 is divisible by 3, omit 9, 3, 6 (8, 1), which are manifestly multiples of 3, and simply add 8 and 7. 223. Other prime factors, 7, 13, 17, 19, sometimes betray their presence to one familiar with the subject; but, practically, the best way to detect them is to attempt to divide by them. 224. If we divide any number less than 121 (112) by 11, or by a number greater than 11, it is plain that the quotient is less than 11. × If we divide any number between 121 and 143 (11 x 13) by 11, the quotient will evidently lie between 11 and 13; and, since there are no prime numbers between 11 and 13, the quotient, if a whole number, must be composite, and contain factors smaller than 11. What is thus proved of 11 and 13 is evidently true of any two adjacent prime numbers; namely, that, excepting the second power of the smaller prime number, every composite number less than the product of two adjacent prime numbers, contains prime factors less than the smaller of these two numbers. Thus, every composite number less than 4087 (61 × 67), except 3721 (613), contains prime factors less than 61. 225. From the preceding article, the value of the following table, in discovering the prime factors of a given number, will be apparent. 101 103 107 109 113 Primes. 79 83 89 97 Opposite to "Powers" are placed the squares of the primes from 7 to 109; and opposite to "Products are placed the products of the successive pairs of adjacent primes from 7 to 113. 226. Find the prime factors of 610,764. As 64 is divisible by 4, but 764 is not divisible 22 610,764 by 8, 22 is the highest power of 2 contained in 3 152,691 7 50,897 11 7,271 610,764. As the sum of the digits 152,691 is divisible by 3 but not by 9, 31 is the highest power of 3 contained in 152,691. The next quotient, 50,897, does not contain 5; but divided by 7 gives 7271. 7271 does not contain 7; but, since 7+7 — (2 + 1) = 11, it is divisible by 11. The quotient 661 when divided by 6 gives a remainder of 1, which shows that it may be a prime number. It cannot be divided by 11, 13, 17, or 19, and is seen by the table to be less than 667 (23 × 29), and not equal to 529 (232); therefore it is a prime number. Thus, 610,764 = 22 × 3 × 7 × 11 × 661. = |