OPERATION. 276)360(1 276 84)276(3 24)84(3 12)24(2 24 0 Hence, to find the greatest common divisor of two numbers, we deduce this RULE. Divide the greater number by the less, then the less number by the remainder; thus continue to divide the last divisor by the last remainder, until there is no remainder. The last divisor will be the greatest common divisor. NOTE.-When there are more than two numbers whose greatest common divisor is required, we must find the greatest common divisor of any two, and then find the greatest common divisor of this divisor thus found, and one of the remaining numbers; and thus continue until all the different numbers have been used. What is the greatest common divisor of two or more numbers? Repeat the rule for finding the greatest common divisor of two numbers. How do you proceed when there are more than two numbers ? EXAMPLES. 1. Find the greatest common divisor of 592 and 999 OPERATION. 592)999(1 592 407)592(1 185)407(2 370 37)185(5 185 0 From which we obtain 37 for the greatest common divisor of 592 and 999. 2. What is the greatest common divisor of 492, 744, and 906? We first find the greatest common divisor of 492 and 744 by the following OPERATION. 492)744(1 492 252)492(1 252 240)252(1 12)240(20 Therefore, the greatest common divisor of 492 and 744 is 12. Again, proceeding with 12 and 906, we have the following OPERATION. 12)906(75 6)12(2 12 0 We thus find 6 to be the greatest common divisor of 12 and 906, and consequently of the three numbers, 492, 744, and 906. 3. What is the greatest common divisor of 315 and 405? Ans. 45. 4. What is the greatest common divisor of 1825 and 2655? Ans. 5. 5. What is the greatest common divisor of 506 and 308? Ans. 22. 6. What is the greatest common divisor of 404 and 364 ? Ans. 4. 7. What is the greatest common divisor of 246, 372, and 522? Ans. 6. 36. We are now prepared to proceed to the reduction of fractions. We know (PROP. VI., ART. 34) that we can divide both numerator and denominator of a fraction by any number without altering its value. If we divide by the greatest common divisor, the resulting fraction will be in its lowest terms. Therefore, to reduce a fraction to its lowest terms, we have this RULE. Divide both numerator and denominator by their greatest common divisor. How do you reduce a fraction to its lowest terms? EXAMPLES. 1. Reduce 2 to its lowest terms. We have already found (Ex. 1, ART. 35,) the greatest common divisor of 592 and 999 to be 37. Dividing both these terms by 37, we find 16 and 27 for quotients: hence, 5, when reduced to its lowest terms, becomes 19. 2. Reduce to its lowest terms. 3. Reduce 30, f, f, to their lowest terms. 4. Reduce 15 to its lowest terms. 5. Reduce to its lowest terms. 6. Reduce 275 to its lowest terms. 7. Reduce, to its lowest terms. 8. Reduce to its lowest terms. 9. Reduce 3523 to its lowest terms. 10. Reduce 58780 to its lowest terms. Ans. . Ans.,,. Ans. 7. Ans. 3. Ans. . Ans. TOT Ans. 1. Ans. t. Ans. . We may frequently discover numbers, by inspection, which will divide both numerator and denominator without a remainder. When this is the case, we need not resort to the rule for obtaining the greatest common divisor, until we have divided by such numbers. 11. Reduce 5184 to its lowest terms. In this example, we first divide the numerator and denominator by 4, which reduces the fraction to 128. We again divide by 4, and obtain 23. Dividing the numerator and denominator of this last fraction by 4, we obtain , which is still further reduced by dividing three successive times by 3. 3. 5184-1228-231=1=37=1=4. 12. Reduce 1 to its lowest terms. ÷2 ÷3÷÷3÷3 ÷3. 122=4=27=&=Z=1. 13. Reduce 98 to its lowest terms. 14. Reduce 1888 to its lowest terms. 15. Reduce 175 to its lowest terms. 16. Reduce 2500 to its lowest terms. Ans. . Ans. H. Ans. Ans.. 37. To reduce an improper fraction to a whole or mixed number. Reduce to a mixed number. Since the value of a fraction is the quotient arising from dividing the numerator by the denominator, (ART. 34,) we may find the value of 15, by dividing 95 by 13. Performing the division, we find 7 for the quotient, and 4 for a remainder. Hence =7†z. (Art. 26.) From which we have the following RULE. Divide the numerator by the denominator; the quotient will be the integral part of the mixed number. The remainder being placed over the denominator of the improper fraction, will form the fractional part. Repeat the rule for reducing an improper fraction to a mixed number. |