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. 105 )35 2, 3, 5, and 7. She was silent the 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 quotient are the prime factors required. ¿OOF.-2 X3 X5X7=210. Hence, 65. 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 remainder, 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 result 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 2×3×7; but it can be divided by no other number. 2. Resolve 4 and 6 into their prime factors. Solution.-4=2×2; and 6=2×3. 3. Resolve 8 into its prime factors. Ans. 8=2×2×2. Resolve the following composite numbers into their prime 76. Resolve 120 and 144 into their prime factors. 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. 4. 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? 163. How do you find a common divisor of two or more numbers? Obs. If the 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. 3. Find a common divisor of 14, 28, 42, and 35. 4. Find a common divisor of 10, 35, 50, 75, and 60. 5. Find a common divisor of 82, 118, and 146. 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 15 common divisor. Hence, 30 170. 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 8. What is the greatest common divisor of 285 and 465 ? 9. What is the greatest common divisor of 532 and 1274? 10. What is the greatest common divisor of 888 and 2775? 11. What is the greatest common divisor of 2145 and 3471 ? 12. What is the greatest common divisor of 1879 and 2425 ? 13. What is the greatest common divisor of 75, 125, and 160? Suggestion.Find the greatest common divisor of 75 and 125, which is 25. Then that of 25 and 160. Ans. 5. 14. What is the greatest common divisor of 183, 3996, 108? 15. What is the greatest common divisor of 672, 1440, and 3472 ? 16. What is the greatest common divisor of 30, 42, and 66 ? Operation. 30=2X3X5. 42=2X3X7 66=2X3X11 Now 2×3=6 Ans. Analysis. By resolving the given numbers into their prime factors, (Art. 165,) we find that the factors 2 and 3 are both common divisors of them. But we have seen that a composite number can be divided by the product of any two or more of its prime factors; (Art. 165. Obs. 2;) consequently 30, 42, and 66 can all be divided by 2×3; for 2×3 is the product of two prime factors common to each. And since they are the only factors common to the given numbers, their product must be the greatest common divisor of them. Hence, .171. Second Method of finding the greatest common divisor of two or more numbers. Resolve the given numbers into their prime factors, and the continued product of those factors which are common to each, will be the greatest common divisor. OBS. If the given numbers have but one common factor, that factor itself is the greatest common divisor. 17. What is the greatest common divisor of 105 and 165 ? 18. What is the greatest common divisor of 36, 60, and 108? 19. What is the greatest common divisor of 108, 126, and 162? 20. What is the greatest common divisor of 105, 210, and 315 ? 21. What is the greatest common divisor of 24, 42, 54, and 60? 22. What is the greatest common divisor of 56, 84, 140, and 168? |