LEAST COMMON MULTIPLE.

172. One number is said to be a multiple of another, when the former can be divided by the latter without a remainder. (Art. 160. Def. 8.) Hence,

173. A common multiple of two or more numbers, is a number which can be divided by each of them without a remainder. Thus, 12 is a common multiple of 2, 3, and 4; 15 is a common multiple of 3 and 5, &c.

OBS. A common multiple is always a composite number, of which each of the given numbers must be a factor; otherwise it could not be divided by them. (Art. 165. Obs. 2.)

174. The continued product of two or more given numbers will always form a common multiple of those numbers. The same numbers may have an unlimited number of common multiples; for, multiplying their continued product by any number, will form a new common multiple. (Art. 161. Prop. 14.)

175. The least common multiple of two or more numbers, is the least number which can be divided by each of them without a remainder. Thus, 12 is the least common multiple of 4 and 6, for it is the least number which can be exactly divided by them.

OBS. The least common multiple of two or more numbers, is evidently composed of all the prime factors of each of the given numbers repeated once, and only once. For, if it did not contain all the prime factors of any one of the given numbers, it could not be divided by that number. (Art. 165. Obs. 2.) On the other hand, if any prime factor is employed more times than it is repeated as a factor in some one of the given numbers, then it would not be the least common multiple.

Ex. 1. What is the least common multiple of 10 and 15?

Analysis.—10=2X5, and 15=3X5. The prime factors of the given numbers are 2, 5, 3, and 5. Now since the factor 5 occurs once in each number, we may therefore cancel it in one

QUEST.-172. When is one number said to be a multiple of another? 173. What is a common multiple? 174. How may a common multiple of two or more numbers be formed? How many common multiples may there be of any given numbers? 175. What is the least common multiple of two or more numbers?

instance, and the continued product of the remaining factors 2X3 X5, or 30, will be the least common multiple.

Operation. 5)10" 15

2 " 3

5×2×3=30 Ans.

We first divide both the numbers by 5 in order to resolve them into prime factors. (Art. 175. Obs.) Thus, all the different factors of which the given num

bers are composed, are found in the divisor and quotients once, and only once. Therefore the product of the divisor and quotients 5X2X3, is the least common multiple required. Hence,

176. To find the least common multiple of two or more numbers.

Write the given numbers in a line with two points between them. Divide by the smallest number which will divide any two or more of them without a remainder, and set the quotients and the undivided numbers in a line below. Divide this line and set down the results as before; thus continue the operation till there are no two numbers which can be divided by any number greater than 1. The continued product of the divisors into the numbers in the last line, will be the least common multiple required.

OBS. 1. We have seen that the least divisor of every number is a prime number; hence, dividing by the smallest number which will divide two or more of the given numbers, is dividing them by a prime number. (Art. 161. Prop. 20.)

The result will evidently be the same, if, instead of dividing by the smallest number, we divide the given numbers by any prime number, that will divide two or more of them, without a remainder.

2. The preceding operation, it will be seen, resolves the given numbers into their prime factors, (Art. 165,) then multiplies all the different factors together, taking each factor as many times in the product, as are equal to the greatest number of times it is found in either of the given numbers.

3. If the given numbers are prime numbers, or are prime to each other, the continued product of the numbers themselves will be their least common multiple. (Art. 168. Obs.) Thus, the least common multiple of 5 and 7 is 35; of 8 and 9 is 72.

QUEST.-176. How is the least common multiple of two or more numbers found? Obs. If the given numbers are prime, or are prime to each other, what is the least common multiple of them? 176. a. Upon what principle does this rule depend? Obs. Why do you divide by the smallest number that will divide two or more of the given numbers without a remainder?

Operation. 6=2X3

Ex. 2. What is the least common multiple of 6, 8, and 12? Analysis. By resolving the given numbers into their prime factors, it will be seen that 2 is found once as a factor in 6; twice in 12; and three times in 8. It must therefore be taken three times in the product. Again, 3 is a fac- 2×2×2×3=24 tor of 6, and 12, consequently it must be taken only once in the product. (Art. 176. Obs. 2.) Thus, 2X2X2X3=24 Ans.

8=2×2×2 12=2×2×3

Ex. 3. What is the least common multiple of 12, 18, and 36?

Explanation. In the first operation, we divide by the smallest numbers which will divide any two or more of the given numbers without a remainder, and the product of the divisors, &c., is 36, which is the answer required.

In the second and third operations, we divide by numbers that will divide two or more of the given numbers without a remainder, and in both cases, obtain erroneous answers.

Note. It will be seen from the second and third operations above, that "dividing by any number, which will divide two or more of the given numbers without a remainder," according to the rule given by some authors, does not always give the least common multiple of the numbers.

176. a. The reason of the preceding rule depends upon the principle that the least common multiple of any two or more numbers, is composed of all the prime factors of the given numbers, each taken as many times, as are equal to the greatest number of times it is found in either of the given numbers. (Art. 175. Obs.)

Nole.-1. The reason for dividing by the smallest number, is because the divisor may otherwise be a composite number, (Art. 161. Prop. 20,) and have a factor common to it and one of the quotients, or undivided numbers in the last line; consequently the continued product of them would be too large for

the least common multiple. (Art. 175. Obs.) Thus, in the second operation the divisor 9, is a composite number, containing the factor 3 common to the 3 in the quotient; consequently the product is three times too large. In the third operation the divisor 12, is a composite number, and contains the factor 6 common to the 6 in the quotient; therefore the product is six times too large.

2. The object of arranging the given numbers in a line, is that all of them may be resolved into their prime factors at the same time; and also to present at a glance the factors which compose the least common multiple required.

4. Find the least common multiple of 6, 9, and 15.

5. Find the least common multiple of 8, 16, 18, and 24. 6. Find the least common multiple of 9, 15, 12, 6, and 5. 7. Find the least common multiple of 5, 10, 8, 18, and 15. 8. Find the least common multiple of 24, 16, 18, and 20. 9. Find the least common multiple of 36, 25, 60, 72, and 35. 10. Find the least common multiple of 42, 12, 84, and 72.` 11. Find the least common multiple of 27, 54, 81, 14, and 63. 12. Find the least common multiple of 7, 11, 13, 3, and 5.

177. The process of finding the least common multiple may often be shortened, by canceling every number which will divide any other given number, without a remainder, and also those which will divide any other number in the same line. The least common multiple of the numbers that remain, will be the answer required.

OBS. By attention and practice, the student will be able to discover, by inspection, the least common multiple of numbers, when they are not large. 13. Find the least common multiple of 4, 6, 10, 8, 12, and 15.

Now, 2X2X3X2X5=120 Ans. proceed with the remaining num

bers as before.

14. Find the least common multiple of 9, 12, 72, 36, and 144. 15. Find the least common multiple of 8, 12, 20, 24, and 25. 16. Find the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9,

17. Find the least common multiple of 63, 12, 84, and 7. 18. Find the least common multiple of 54, 81, 63, and 14. 19. Find the least common multiple of 72, 120, 180, 24, and 36.

177. a. The least common multiple of two or more numbers may also be found in the following manner.

First find the greatest common divisor of two of the given numbers; by this divide one of these two numbers, and multiply the quotient by the other. Then perform a similar operation on the product and another of the given numbers; thus continue the process until all of the given numbers have been employed, and the final result will be the least common multiple required.

20. What is the least common multiple of 24, 16, and 12? Solution. By inspection, we find the greatest common divisor of 24 and 16, is 8. Now 24÷8=3; and 3×16=48. Again, the greatest common divisor of 48 and 12, is 12. Now 48-12

-4; and 4X12=48. Ans.

PROOF.-Resolving the given numbers into their prime factors, 24=2×2×2×3; 16=2×2×2×2; and 12=2×2×3; (Art. 165;) consequently, 2×2×2×2×3=48, the least common multiple. (Art. 175. Obs.)

OBS. The reason of this rule depends upon the principle, that if the product of any two numbers be divided by any factor which is common to both, the quotient will be a common multiple of the two numbers. Thus, if 48, the product of 6 and 8, be divided by 2, a factor of both, the quotient 24, will be a multiple of each, since it may be regarded either as 8 multiplied by the quotient of 6 by the factor 2, or as 6 multiplied by the quotient of 8 by the same factor. Hence, it is obvious, that the greater the common measure is, the less will be the multiple; and, consequently, the greatest common measure will produce the least common multiple.

When the common multiple of the first two numbers is found, it is evident, that any number which is a common multiple of it and the third number, will be a multiple of the first, second, and third numbers.

21. What is the least common multiple of 75, 120, and 300? 22. What is the least common multiple of 96, 144, and 720? 23. What is the least common multiple of 256, 512, and 1728? 24. What is the least common multiple of 375, 850, and 3400 ?