NOTE. We may also define thus: a multiple of a number is another number which will exactly contain the former; or, a multiple of a number is a number of which the former is an exact divisor, etc. The least common multiple may be represented by the initials L. C. M. PRINCIPLES. 1. A multiple of a number is exactly divisible by that number. 2. A multiple of a number must contain all the prime factors of that number. 3. A common multiple of two or more numbers must contain all the prime factors of each of those numbers. 4. The least common multiple of two or more numbers must contain all the prime factors of each number, and no other factors. CASE I. 184. When the numbers are small and can be readily factored. 185. The First Method consists in resolving the numbers into their prime factors, and taking the product of all the different factors. OPERATION. 20=2×2×5 70=2×5×7 2×2×5×3×7=420. 1. Find the least common multiple of 20, 30, and 70. SOLUTION. We first resolve the numbers into their prime factors. A multiple of 20 must contain the factors of 20, or 2, 2, 5; a multiple of 30 must contain the factors of 30, or 2, 3, 5; a multiple of 70 must contain the factors of 70, or 2, 5, 7; hence the least common multiple of 20, 30, and 70 must contain all these different factors and no others; therefore 2×2×5×3×7, or 420, is the L. C. M. of 20, 30, and 70 (Prin. 4). Rule. I. Resolve the numbers into their prime factors. II. Take the product of all the different factors, using each factor the greatest number of times it occurs in either number. NOTE. Any numbers which are divisors of the others may be omitted, since the multiple of the other numbers will be a multiple of these. Find the least common multiple of 2. 25, 30, and 42. 3. 11, 32, and 40. 4. 56, 72, and 96. Ans. 1050. Ans. 1760. Ans. 2016. 186. The Second Method consists in taking out the prime factors of the least common multiple, and finding their product. 1. Find the least common multiple of 24, 30, and 70. OPERATION. 3 12-15-35 5 4-5-35 4-1-7 2×3×5x4x7=840 SOLUTION.-Placing the numbers one beside another, and dividing by 2, we find that 2 is a factor of each of them; it is therefore a factor of the L. C. M. (Prin. 4); dividing the quotients by 3, we find that 3 is a factor of some of the numbers; it is therefore a factor of the least common multiple (Prin. 4); dividing the next quotients by 5, we find that 5 is a factor of some of the numbers; it is therefore a factor of the L. C. M.; and the quotients having no other common factors, we see that all the different factors of the given numbers are 2, 3, 5, 4, and 7; hence their product, which is 840, is the L. C. M. required. Hence the following Rule.-I. Write the numbers one beside another, divide by any prime number that will exactly divide two or more, and write the quotients and undivided numbers beneath. II. Divide the quotients in the same manner, and thus continue until no two numbers in the lowest line have a common factor. III. Take the product of the divisors and final quotients; the result will be the least common multiple required. Find the least common multiple of 2. 12, 18, 24, and 27. 3. 22, 33, 55, and 66. 4. 14, 19, 38, and 57. 5. 64, 84, 120, and 216. 6. 1, 2, 3, 4, 5, 6, 7, and 8. 7. 18, 36, 126, 40, and 48. 8. 13, 37, 7, 91, and 11. 10. 15, 25, 45, 75, 135, and 209 Ans. 216. Ans. 330. Ans. 798. Ans. 60480. Ans. 840. Ans. 5040. Ans. 37037. Ans. 10080. Ans. 141075. CASE II. 187. When the numbers are large and cannot be readily factored. 1. Find the least common multiple of 45 and 72. OPERATION. 45=5X9 SOLUTION.-The greatest common divisor of these numbers is 9; 45 equals 5 times 9 and 72 equals 8 times 9; hence the L. C. M., as found in the first method, is 5× 9 × 8, which equals 45x22; and which we see is the first number multiplied by the second divided by their greatest common divisor. From the result of this operation we may derive the following rule: L. C. M.=5X9X8 Rule.-I. Find the greatest common divisor of two numbers, divide one number by it, and multiply the other number by the quotient. II. When there are more than two numbers, find the least common multiple of two of the numbers, and then of this number and the third number, etc. Find the least common multiple of 1. What is the least sum of money with which I could purchase either pigs at $5, sheep at $7, cows at $40, or horses at $75? Ans. $4200. 2. A can dig 5 rods of ditch in a day, B 8 rods, C 12 rods, and D 15 rods; how many rods will it require to give an exact number of days' work for each? Ans. 120 rods. 3. The circumferences of the driving wheels of four locomotives are 12, 18, 20, and 21 feet respectively; what is the shortest distance in which each wheel can make an exact number of revolutions? Ans. 1260 ft. 4. A, B, C, and D start from the same point; A goes a mile in 15 minutes, B in 18 minutes, C in 21 minutes, and D in 25 minutes; how far can each travel, and all return to the starting point at the same time? Ans. A, 210; etc. 5. There is an island 120 miles in circumference, and A, B, C, and D start to travel around it; A goes 12 miles a day, B 15, C 20, and D 24; in what time would they all come together at the starting point? Ans. 120 days. 6. The circumferences of the wheels of a carriage are 13 and 16 feet respectively; a nail on the tire of each was on top when the carriage started; how far will the carriage have gone when the nails shall be uppermost for the 450th time, there being 5280 feet in a mile? Ans. 17 mi. 3840 ft. ABBREVIATED METHOD. 188. The Abbreviated Method of Least Common Multiple here given will be found useful in practice. OPERATION. 45-48-80-1120 3-2-12 3-1 1. Find the least common multiple of 45, 48, 80, and 120. SOLUTION. Having written the numbers in a line as in the last method, we cut off 120, the left hand number. Now, the least common multiple must consist of 120, multiplied by those factors of the other numbers which are not found in 120; dividing 45 by 15, the greatest divisor common to it and 120, we obtain 3; dividing 48 by 24, the greatest divisor common to it and 120, we have 2; dividing 80 in the same way, we have 2. As the factors thus obtained are not prime to each other, we cut off 2, and divide by the greatest divisors common to it and 3 and 2 respectively, when we have 3 and 2 as the only factors of the L. C. M. not contained in 120. Multiplying 120 by these factors, we have 720 for the L. C. M. Hence the following Rule.-I. Write the numbers one beside another, cut off any convenient number, generally the largest, and draw a line beneath them. II. Divide each remaining number by the greatest divisor common to it and the number cut off. If the factors thus obtained are not prime to each other, cut off one of them and proceed as before until all the factors are prime to each other. III. Multiply the numbers cut off and the last row of factors together for the least common multiple. NOTES.-1. If one number is contained in any of the others, it may be omitted. 2. If the number cut off is found to be prime to the others in the same line, cut off another and proceed as before, reserving the first as a factor of the L. C. M. Find the least common multiple of 2. 30, 80, 120, and 135. 3. 77, 91, 143, and 165. Ans. 2160. Ans. 15015. Ans. 237336. 4. 93, 132, 232, and 319. GENERAL PRINCIPLES OF GREATEST COMMON DIVISOR AND LEAST COMMON MULTIPLE. 189. These General Principles express the relations between the greatest common divisor and the least common multiple. 1. The greatest common divisor of two or more numbers is a divisor of their least common multiple. 2. The product of two numbers divided by their greatest common divisor equals their least common multiple. 3. The product of the relatively prime parts of two or more numbers multiplied by the G. C. D. equals the L. C. M. 4. The quotient of the L. C. M. of two or more numbers, divided by their G. C. D., equals the product of the factors not common. 5. The prime factors not common may be found by resolving the quotient of the L. C. M. divided by the G. C. D. into its prime factors. 6. The G. C. D., multiplied by each of the factors not common, will give numbers having the same G. C. D. and L. C. M. NOTE. The pupil may be required to illustrate these principles. PRACTICAL PROBLEMS. 1. The L. C. M. of 6 and 8 and a number prime to each of them is 120; what is the third number? SOLUTION.-120 contains all the factors of 6, of 8, and of the 3d number; hence all the factors of 120 not found in 6 and 8 constitute the third number. The only factor is 5, therefore 5 is the number required. OPERATION. 120 23x3x5 .. 5 the number. |