96 PROPERTIES OF NUMBERS. 10. Change 35261 from the decimal to the 11. Change 643175 from the decimal to the 12. Change 175683 from the decimal to the 13. Change 534610 from the decimal to the 14. Change 841568 from the decimal to the 15. Change 592835 from the decimal to the Nete. Since every scale requires as many characters the radix, we will denote 10 by t, and 11 by e. Ans. 247 163. To change a number expressed in a notation, to the decimal scale. Multiply the left hand figure by the given r product add the next figure; then multiply this again, and to this product add the next figure; operation till all the figures in the given number ha and the last product will be the number in the deci 16. Change 3204 from the quinary to the deci Explanation. Multiplying the left hand figure by 5, the given radix, evidently reduces it to the next lower order; for in the quinary scale, 5 in an inferior order make one in the next superior order. For the same reason, multiplying this sum by 5 again, reduces it to the next lower order, &c. OBS. This and the preceding operations are the same in prin compound numbers from one denomination to another. 17. Change 1322232 from the quaternary to the 18. Change 2546571 from the octary to the deci 19. Change 34120521 from the senary to the de 20. Change 145620314 from the septenary to the 21. Change 834107621 from the nonary to the de 22. Change 403130021 from the quinary to the d 23. Change 704400316 from the octary to the de 24. Change 903124106 from the duodecimal to the d ANALYSIS OF COMPOSITE NUMBERS. 164. Every composite number, it has been shown, may be resolved into prime factors. (Art. 161. Prop. 19.) Ex. 1. Resolve 210 into its prime factors. Operation. 2)210 We first divide the given number by 2, which is the least number that will divide it without a remainder, and which is also a prime number. (Prop. 20.) We next divide by 3, then by 5. The several divisors and the last Ans. 2, 3, 5, and 7. quotient are the prime factors required. 3)105 5)35 7 165. To resolve a composite number into its prime factors. Divide the given number by the smallest number which will divide it without a remainder; then divide the quotient in the same way, and thus continue the operation till a quotient is obtained which can be divided by no number greater than 1. The several divisors with the last quotient, will be the prime factors required. (Art. 161. Prop. 19.) Demonstration. Every division of a number, it is plain, resolves it into two factors, viz: the divisor and dividend. (Art. 112.) But according to the rule, the divisors, in every case, are the smallest numbers that will divide the given number and the successive quotients without a remainder; consequently they are all prime numbers. (Art. 161. Prop. 20.) And since the division is continued till a quotient is obtained, which cannot be divided by any number greater than 1, it follows that the last quotient must also be a prime number; for, a prime number is one which cannot be exactly divided by any whole number except a unit and itself. (Art. 160 Def. 4.) OBS. 1. Since the least divisor of every number is a prime number, it is evident that a composite number may be resolved into its prime factors, by dividing it continually by any prime number that will divide the given number and the quotients without a remainder. Hence, 2. A composite number can be divided by any of its prime factors without a zemainder, and by the product of any two or more of them, but by no other number. Thus, the prime factors of 42 are 2, 3, and 7. Now 42 can be di QUEST.-165. How do you resolve a composite number into its prime factors? Obs. Will the same resuit be obtained, if we divide by any of its prime factors? vided by 2, 3, and 7; also by 2×3, 2×7, 3×7, and 2) divided by no other number. 2. Resolve 4 and 6 into their prime factor. Solution.-4=2×2; and 6=2×3. Ans. 8= 3. Resolve 8 into its prime factors. Resolve the following composite numbers 76. Resolve 120 and 144 into their prime factors 77. Resolve 180 and 420 into their prime factors 78. Resolve 714 and 836 into their prime factors. 79. Resolve 574 and 2898 into their prime factor 80. Resolve 11492 and 180 into their prime facto 81. What are the prime factors of 650 and 1728? 82. What are the prime factors of 1492 and 8032 83. What are the prime factors of 4604 and 1680 84. What are the prime factors of 71640 and 2032 85. What are the prime factors of 84705 and 6594 86. What are the prime factors of 9235% and 8137 GREATEST COMMON DIVISOR. 166. A common divisor of two or more numbers, is a number which will divide each of them without a remainder. Thus 2 is a common divisor of 6, 8, 12, 16, 18, &c. 167. The greatest common divisor of two or more numbers, is the greatest number which will divide them without a remainder. Thus 6 is the greatest common divisor of 12, 18, 24, and 30. OBS. A common divisor is sometimes called a common measure. It will be seen that a common divisor of two or more numbers, is simply a factor which is common to those numbers, and the greatest common divisor is the greatest factor common to them. Hence, 168. To find a common divisor of two or more numbers. Resolve each number into two or more factors, one of which shall be common to all the given numbers. Or, resolve the given numbers into their prime factors, then if the same factor is found in each, it will be a common divisor. (Art. 165. Obs. 2.) OBS. If the given numbers have not a common factor, they cannot have a common divisor greater than a unit; consequently they are either prime numbers, or are prime to each other. (Art. 160. Def. 3. Obs. 2.) Note.-The following facts may assist the learner in finding common divisors: 1. Any number ending in 0, or an even number, as 2, 4, 6, &c., may be divided by 2. 2. Any number ending in 5 or 0, may be divided by 5. 3. Any number ending in 0, may be divided by 10. 4. When the two right hand figures are divisible by 4, the whole number may be divided by 4. 5. If the three right hand figures of any number are divisible by 8, the whole is divisible by 8. Ex. 1. Find a common divisor of 6, 15, and 21. Solution.-6=3X2; 15=3X5; and 21=3X7. The factor 3 is common to each of the given numbers, and is therefore a common divisor of them. QUEST.-166. What is a common divisor of two or more numbers? 167. What is the greatest common divisor of two or more numbers? Obs. What is a common divisor sometimes called? 168. How do you find a common divisor of two or more numbers? Obs. If two given numbers have not a common factor, what is true as to a common divisor? 2. Find a common divisor of 15, 18, 24, and 36. 6. Find a common divisor of 42 and 66. Ans. 2, 3, or 6. 169. It will be seen from the last example that two numbers may have more than one common divisor. In many cases it is highly important to find the greatest divisor that will divide two or more given numbers without a remainder. 7. What is the greatest common divisor of 35 and 50? Operation. 35)50(1 35 15)35(2 Dividing 50 by 35, the remainder is 15, then dividing 35 (the preceding divisor) by 15 (the last remainder) the remainder is 5; finally, dividing 15 (the preceding divisor) by 5 (the last remainder) nothing remains; con5)15(3 sequently 5, the last divisor, is the greatest common divisor. Hence, 30 15 170. 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. When there are more than two numbers given. First find the greatest common divisor of any two of them; then, that of the common divisor thus obtained and of another given number, and so on through all the given numbers. The last common divisor found, will be the one required. Demonstration.-Since 5 is a measure of the last dividend 15, in the preceding solution, it must therefore be a measure of the preceding dividend 35; because 35-2X15+5; and 35 is one of the given numbers. Now, since 5 measures 15 and 35, it must also measure their sum, viz: 35+15, or 50, which Is the other given number. (Art. 161. Prop. 13.) In a similar manner it may be shown that the last divisor will, in all cases, be the greatest common divisor. Note.-Numbers which have no common measure greater than 1, are said to be incommensurable. Thus 17 and 29 are incommensurable. QUEST.-170. How find the greatest common divisor of two numbers? Of more than two? |