least common denominator of fractions, it becomes necessary to show how to find THE LEAST COMMON MULTIPLE. A multiple of several numbers is such a number as can be divided by each of them without a remainder. Thus, 12, 24, 36, 48, &c., are multiples of 2, 3, 4, and 6, since each of them is divisible by 2, 3, 4, and 6. Any set of numbers may have an infinite number of multiples. practice it is the least common multiple which is usually sought. In the above example, 12 is the least common multiple of 2, 3, 4, and 6. In Let us seek the least common multiple of two numbers, as for example, of 4 and 18. Separating these numbers into their smallest component parts, (ART. 21,) they become 4=2×2; 18=2×3×3. If we multiply 2×2=4 by 2×3×3=18, we shall obtain 2×2×2×3×3, which is obviously a common multiple of 4 and 18, since the factors of these numbers are found in this expression. But it is not the least common multiple of 4 and 18, since one of the 2's, which is a common factor of 4 and 18, may be omitted, and the result, 2×2×3×3, will still contain all the different factors of 4 and 18. Hence, when two numbers have no common divisor, their least common multiple may be found by taking their product. When they have a common divisor, their least common multiple may be found by dividing their product by their greatest common divisor; or, by dividing one of the numbers by their greatest common divisor, and multiplying the quotient by the other number; or, by dividing each number by their greatest common divisor, and multiplying the product of the quotients by this greatest common divisor, It is the last method that we find most convenient to employ. The least common multiple of more than two numbers may be found, by first finding the least common multiple of any two of the numbers, and then finding the least common multiple of that multiple, and another of the given numbers, and so on, until all the different numbers have been used. We will now seek the least common multiple of 10, 18 and 21. The greatest common divisor of 10 and 18 is 2. Dividing 10 and 18 by 2, we find 5 and 9 for the quotients; hence the least common multiple of 10 and 18 is 2×5×9. We now seek the least common multiple of 2×5×9 and 21, The greatest common divisor of these two numbers is 3, it being a divisor of 9 and of 21. Dividing by 3, we have 2×5×3 and 7 for the quotients, Hence the least common multiple of 2×5x9=90 and 21, is 3×2×5×3×7=630, which is also the least common multiple of 10, 18 and 21, If we place the numbers 10, 18 and 21 in a horizontal line, and divide the 10 and 18 by 2, and bring down the 21, we shall obtain a second line consisting of 5, 9, and 21. Dividing the 9 and 21 of this 2 | 10, 18, 21 3 5, 9, 21 5, 3, 7 2x3x5x3x7=630, second line by 3, we obtain a third line consisting of 5, 3, and 7, no two of which have a common divisor. Now, multiplying the divisors 2 and 3 by the product of the numbers in the last horizontal line, we have 630, the least common multiple sought. Hence the least common multiple of any set of numbers may be found by the following RULE. Write the numbers in a horizontal line; divide them by the least number which will divide two or more of them without a remainder; place the quotients with the undivided numbers, if any, for a second horizontal line; proceed with this second line as with the first; and so continue until there are no two numbers which can be exactly divided by the same divisor. The continued product of the divisors, and of the numbers in the last horizontal line, will give the least common multiple. NOTE.-When there is no number which will divide two of the given numbers, their continued product must be taken for the least common multiple. What is a multiple of several numbers? Mention some of the multiples of 2, 3, 4, and 6. Are the number of multiples of any set of numbers limited? Repeat the Rule for finding the least common multiple of any set of numbers. When there is no number which will divide two of the given numbers, how is the least multiple found? EXAMPLES. 1. What is the least common multiple of 12, 16, and 24? Hence, 2×2×2×3×2-48 is the least common multiple. 2. What is the least common multiple of 12, 15, 24? Therefore, 2×2×3×5×2=120 is the multiple sought. 3. What is the least common multiple of 11, 77, 88? Ans. 616. 4. What is the least common multiple of 37, 41 ? Ans. 1517. 5. What is the least common multiple of 24, 60, 45, 180? Ans. 360. 6. What is the least common multiple of 2, 4, 6, 8? Ans, 24. 7. What is the least common multiple of 3, 5, 7, 9? Ans. 315. 8. What is the least common multiple of 2, 3, 4, 5, 6, 7, 8,9? Ans. 2520. 9. What is the least common multiple of 7, 14, 16, 18, 24? Ans. 1008. 10. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9, 11? Ans. 27720. 42. We are now prepared to reduce fractions to their least common denominator. Let it be required to reduce to the least common denominator the fractions F, 6, and 11. If we take the least common multiple of the denominators 12, 16, and 24, which is 48, and divide it in turn by these denominators, we shall obtain the respective quotients 4, 3, and 2. Hence, if we multiply the numerator and denominator of each fraction by 4, 3 and 2 respectively, they will become 48, 4 and . These fractions are equivalent to the original ones, and have their least common denominator. Hence fractions may be reduced to their least common denominator by the following RULE. Reduce the fractions to their simplest form; then find the least common multiple of their denominators, (by Rule under ART. 41,) which will be their least common denominator. Divide this denominator by the respective denominators of the given fractions; multiply the quotients thus obtained by the respective numerators, and the several products will be the new numerators. Repeat the Rule for reducing fractions to their least common denominator. EXAMPLES. 1. Reduce 2, 5, 4, to equivalent fractions having the least common denominator. The least common multiple of the denominators 12, 15, 24, is 120 common denominator. = New numerator of first fraction 120 x 5=50. 24 |