46.* How many balls at 15 pence apiece, can be bought for 146 pence? Operation. 3)149 5)49, 2 rem. 9, 4 rem. Ans. 9 and 14r. Analysis. This divisor being composite, whose factors are 3 and 5, we first divide by 3, and this quotient by 5. To find the true remainder, multiply each remainder by all the divisors preceding the division from which it arose, and to the sum of the products add the first remainder; the result will be the true remainder. In this example the only preceding divisor is 3; now the last remainder 4×3=12, and 12+2=14 the true remainder. Hence, 66. To divide by a composite number. Divide the dividend by one of the factors, then divide the quotient thus obtained by another factor, and so on till all the factors are employed. The last quotient will be the answer. 47. Divide 231 by 21, using its factors. 48. Divide 195 by 16, using its factors. 49. Divide 256 by 24, using its factors. 50. Divide 365 by 48, using its factors. 51. Divide 410 by 45, using its factors. 52. Divide 217 by 63, using its factors. 53. Divide 561 by 56, using its factors. 54. Divide 893 by 72, using its factors. 55. Divide 1275 by 96, using its factors. Ans. 11. 67. To divide by 10, 100, 1000, &c. Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor. The remaining figures of the dividend will be the quotient, and those cut off the remainder. 56. Divide 1325 by 10. 58. Divide 5633 by 1000. 57. Divide 4626 by 100. 59. Divide 8465 by 1000. 60. Divide 26244 by 1000. 61. Divide 136056 by 10000. 62. Divide 2443667 by 100000. 63. Divide 23454631 by 1000000. 68. When there are ciphers on the right hand of the divisor. Cut off the ciphers from the divisor; also cut off as many figures from the right of the dividend. Then divide the remaining figures of the dividend by the remaining figures of the divisor, and the result will be the quotient. Finally, annex the figures cut off from the dividend to the remainder, and the number thus formed will be the true remainder. 64. At 200 dollars apiece, how many carriages can be bought for 4765 dollars? Having cut off the two ciphers on the right of the divisor, and two figures on the right of the dividend, we divide the 47 by 2 in the usual way. 65. Divide 2658 by 20. 66. 3642 by 30. 72. 23148 by 1200. Operation. 200)47 65 Ans. 23-165 rem. Ans. 132 and 18 rem., or 13218. 74. 50382 by 1800. 84. 6075071623. 86. 4367238-2367. 79. 9000000 by 300000. 81. 467342-341. 83. 506725-603. 85. 736241-2764. 87. 6203451-3827. 88. 8230732-3478. 90. 93670858÷ 67213. 89. 8235762-42316. 91. 98765421÷84327. QUEST.--68. When there are ciphers on the right of the divisor, how pro ceed! What is to be done with figures cut off from the dividend? ARITHMETICAL TERMS. 71. Numbers are divided into abstract and concrete. 1. Abstract numbers are numbers used without application to any object; as two, three, four, five, &c. 2. Concrete numbers are numbers applied to some particular object; as two peaches, three pounds, &c. 3. Numbers are also divided into prime and composite. 4. A prime number is one which cannot be produced by multiplying any two or more numbers together; or which cannot be exactly divided by any whole number, except a unit and itself. Thus, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, &c., are prime numbers. OBS. 1. The least divisor of every number is a prime number. 2. One number is said to be prime to another, when a unit is the only number by which both can be divided without a remainder. 3. The number of prime numbers is unlimited. All below fifty are given above. The pupil can easily point out others. 5. A composite number is one which is produced by multiplying two or more factors together. Thus, 12=4 × 3. 6. An even number is one which can be divided by 2 without a remainder; as, 4, 6, 8, 10. 7. An odd number is one which cannot be divided by 2 without a remainder; as, 1, 3, 5, 7, 9, 15. 8. One number is a measure of another, when the former will divide the latter, without a remainder. Thus 2 is a measure of 4; 3 is a measure of 6. 9. A common measure is a number, which will divide two or more numbers, without a remainder. Thus, 2 is a common measure of 4, 6, and 8. 10. The aliquot parts of a number are the parts by which it can be divided without a remainder. Thus, 3 and 7 are aliquot parts of 21. QUEST.-71. What are abstract numbers? Concrete? Prime? Composite? An even number? Au odd number? A common measure? GREATEST COMMON DIVISOR. · 72. A Common Divisor is a number which will di vide two or more numbers without a remainder. 2 is a common divisor of 4, 6, 8, 12, 16. Thus, 73. The Greatest Common Divisor of two or more numbers, is the greatest number which will divide each of them without a remainder. Thus, 6 is the greatest common divisor of 12, 18, and 24. Operation. 30)42(1 30 1. What is the greatest common divisor of 30 and 42 ? Suggestion.-Dividing 42 by 30, the remainder is 12; then dividing 30 (the preceding divisor) by 12 (the last remainder) the remainder is 6; finally, dividing 12 (the preceding divisor) by 6 (the last remainder) nothing remains; consequently 6, the last divisor, is the greatest common divisor. Hence, 12)30(2 24 6)12(2 12 74. To find the greatest common divisor of two numbers. Divide the greater number by the less; then divide the preceding divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor. 2. What is the greatest com. divisor of 56 and 140 ? 3. What is the greatest com. divisor of 116 and 203? 4. Find the greatest common divisor of 148 and 185. 5. Find the greatest common divisor of 237 and 395. 6. What is the greatest com. divisor of 122 and 427? 75. To find the greatest common divisor of more than two numbers? First find the greatest common divisor of any two of QUEST.-What are aliquot parts of a number? 72. A common divisor? 73. The greatest common divisor of two or more rumbers? 74. How find the greatest common divisor of two numbers? 75. Of more than two? the given numbers; then, that of the common divisor thus obtained and of another number, and so on through all the given numbers. The last common divisor found, will be the one required. 7. Find the greatest com. divisor of 45, 63 and 108. 8. Find the greatest com. divisor of 32, 48 and 200. 9. Find the greatest com. divisor of 256, 372 and 522. LEAST COMMON MULTIPLE. 76. A multiple is a number which can be divided by another number without a remainder. Thus, 4 is a multiple of 2; 10 is a multiple of 5. 77. A common multiple is a number which can be divided by two or more numbers without a remainder. Thus, 12 is a common multiple of 2, 3, and 4. 78. The continued product of two or more numbers is always a common multiple of those numbers. 79. The least common multiple of two or more numbers, is the least number which can be divided by each of them without a remainder. Thus, 12 is the least common multiple of 4 and 6. 10. Find the least common multiple of 6, 10, and 12. Suggestion.-Write the given num Operation. 2)6'10"12 3)3" 5" 6 1" 5" 2 bers in a line, and, dividing by 2 the smallest number that will divide any two of them without a remainder, set the quotients 3, 5, and 6 in a line below. Next divide this line by 3 and set the quotients and undivided number 5 in a line as before. Finally, multiply all the divisors into the quo 2 × 3 × 5 × 2=60 QUEST.-76. What is a multiple? 77. A common multiple? 79. What is the least common multiple of two or more numbers? |