LEAST COMMON MULTIPLE.

201. A common multiple of two or more numbers is a number that can be divided by each of them without a remainder; thus, 14 is a common multiple of 2 and 7.

The least common multiple of two or more numbers is the least number that can be divided by each of them without a remainder; thus, 12 is the least common multiple of 4 and 6.

202. A multiple of a number contains all the prime factors of that number; the common multiple of two or more numbers contains all the prime factors of each of the numbers; and the least common multiple of two or more numbers contains only each prime factor taken the greatest number of times it is found in any of the several numbers. Hence,

1. The least common multiple of two or more numbers must be the least number that will contain all the prime factors of them, and none others.

2. The least common multiple of two or more numbers, which are prime to each other, must equal their product.

3. The least common multiple of two or more numbers must equal the product of their greatest common divisor, by the factors of each number not common to all the numbers.

4. The least common multiple of two or more numbers, divided by any one of them, must equal the product of those factors of the others not common to the divisor.

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

Ex. 1. What is the least common multiple of 8, 16, 24, 32, 44.

Ans. 1056. Resolving the

numbers into their prime factors, we find their different prime factors to be 2, 3, and 11. The greatest

1056 Ans. number of times

the 2 occurs as a

factor in any of the given numbers is 5 times; the greatest num

ber of times 3 occurs in any of the numbers is once; and the greatest number of times the 11 occurs in any of the numbers is once. Hence, 2, 2, 2, 2, 2, 3, and 11 must be all the prime factors necessary in composing 8, 16, 24, 32, and 44; and consequently, 1056, the product of these factors, is the least common multiple required (Art. 202).

the divisor and remainders are all prime to each other. Then, these, since they include all the factors necessary to form the given numbers and no others, we multiply together for the required least common multiple, and obtain 1056, as before.

The least common multiple of two or more numbers may be found generally by a process much shorter than either of the above methods, by canceling any number that is a factor of any other of the given numbers, and also by dividing the numbers by such a composite number as may be observed to be their common or greatest common divisor.

Thus, in the third operation, 8 being a factor of several of the numbers, and 16 being a factor of one other number, we cancel them; and observing that 4 is the greatest common divisor of the remaining numbers, we divide them by it.

We next divide by 2, as in the second operation. The numbers in the lower line then being prime to each other, we multiply them and the divisors together, and obtain 1056 as the least common multiple.

FOURTH OPERATION.

8 16 24 32 44
4 11
1056 Ans.

24 X 4 X 11

The fourth operation exhibits a process yet more contracted. The 8 and 16 being factors each of one or more of the other numbers, we cancel them, as in the third operation. Of the remaining numbers we cut off 24 by a short vertical line from the rest as a factor of the least common multiple sought. We then strike out of the two remaining numbers the largest factor each has in common with the 24, by dividing each of them by the greatest common divisor

between it and 24, and write the result beneath. The numbers in the lower line having no factor in common, we carry the process no further. The continued product of the number cut off by the numbers in the lower line gives 1056, the least common multiple, as by the other methods. In this instance we cut off the 24, but either the 32 could have been separated from the rest, or the 44 cut off, and the needless factors striken out with like result. If, however, we had cut off the 44, the numbers placed in the second line would have contained factors common to each other, so that it would have been necessary in that line to have cut off and stricken out factors as before. The reason for this abridged process is, that by the separating off, and by the striking out of factors, we get rid, in an expeditious way, of the factors not required to form the least common multiple sought.

RULE 1. - Resolve the given numbers into their prime factors. The product of these factors, taking each factor only the greatest number of times it occurs in any of the numbers, will be the least common multiple.

Or,

RULE 2.- Having arranged the numbers on a horizontal line, cancel such of them as are factors of any of the others, and separate some convenient one from the rest. Reject from each of the numbers remaining the greatest factor common to it and that number, and write the result in a line below. Should there be in the second line numbers having factors in common, proceed as before; and so continue until the numbers written below are prime to each other. The continued product of the number or numbers separated from the others with those in the last line will be the least common multiple.

NOTE 1. Some give a preference to the following rule for finding the least common multiple: Having arranged the numbers on a horizontal line, divide by such a prime number as will exactly divide two or more of them, and write the quotients and undivided numbers in a line beneath. So continue to divide until the quotients shall be prime to each other. Then the product of the divisors and the numbers of the last line will be the least common multiple.

NOTE 2.-The least common multiple of two or more numbers that are prime to each other is found by multiplying them together (Art. 202).

NOTE 3. - When a single number alone is prime to all the rest, it may be separated off, and used only as a factor of the least common multiple sought. NOTE 4. When the least common multiple of several numbers, and all the numbers except one, which is prime to the others, are given, to find the unknown number, divide the least common multiple given by that of the known numbers (Art. 202).

EXAMPLES.

2. What is the least common multiple of 3, 13, 37, and 91.

3. What is the least common multiple of 9, 14, 30, 35, and 47 ?

Ans. 29610.

4. What is the least common multiple of 6, 8, 10, 18, 20, and 24?

5. What is the least common multiple of 14, 19, 38, and 57 ? Ans. 798. 6. What is the least common multiple of 20, 36, 48, and 50? Ans. 3600. 7. What is the least common multiple of 15, 25, 35, 45, and 100? Ans. 6300. 8. What is the least common multiple of 100, 200, 300, 400, and 575?

Ans. 27600.
3, 4, 5, 6, 8, 9, and
What is that other
Ans. 7.

9. The least common multiple of 1, 2, one other number prime to them, is 2520. number? 10. What is the least common multiple of 18, 24, 36, 126, 20, and 48?

11. I have four different measures; the first contains 4 quarts, the second 6 quarts, the third 10 quarts, and the fourth 12 quarts. How large is a vessel, that may be filled by each one of these, taken a certain number of times full? Ans. 60 quarts.

12. What is the smallest sum of money with which I can purchase a number of oxen at $50 each, cows at $ 40 each, or horses at $75 each? Ans. $600.

MISCELLANEOUS EXAMPLES.

1. How many times does 7 occur as a factor of 6174?

Ans. 3 times.

2. Required the largest prime factor of 5775.
3. Required the largest composite factor of 19929.

Ans. 6643.

4. Required the quotients of 2338 divided by its two prime factors next larger than 1.

Ans. 1169; 334.

5. Required all the prime numbers that will divide 17385 without a remainder.

6. A farmer has 3000 bushels of grain; which are the three smallest-sized bags, and the three largest-sized bins, holding an exact number of bushels, that will each measure the same without a remainder?

Ans. Bags of 1, 2, or 3 bushels each; and bins of 1500, 1000, or 750 bushels each.

7. A teacher having a school consisting of 152 ladies and 136 gentlemen, divided it in such a manner that each class of ladies equalled each class of gentlemen, and the classes were the largest the school would admit of, and have them all of the same size. Required the number of classes, and the number in each class.

Ans. 19 classes of ladies, 17 classes of gentlemen, and 8 pupils in a class.

8. At noon the second, minute, and hour hand of a clock are together; how long after will they be again, for the first time, in the same position?

rails that can be used in

9. J. Porter has a four-sided garden, the first side of which is 348 feet in length; the second, 372 feet; the third, 444 feet; and the fourth, 492 feet. Required the length of the longest fencing it, allowing the end of each rail to lap by the other 9 inches, and all the panels to be of equal length; also, the number of rails, if 5 rails be allowed to each panel. Ans. Length 12ft. 9in.; and 690 rails.

10. L. Ford has 5 pieces of land, the first containing 3A. 2R. 1p.; the second, 5A. 3R. 15p.; the third, 8A. 29p.; the fourth, 12A. 3R. 17p.; and the fifth, 15A. 31p. Required the largest sized house-lots, containing each an exact number of square rods, into which the whole can be divided.

Ans. 1A. 27p. each. 11.° What three numbers between 30 and 140 have 12 for their greatest common divisor, and 2772 for their least common multiple. Ans. 36, 84, and 132.

12. Four men, A, B, C, and D, are engaged in making regular excursions into the country, between which each stays at home just 1 day; and A is always absent exactly 3 days, B 5 days, and C and D 7 days. Provided they all start off on the same day, how many days must elapse before they can all be at home again on the same day? Ans. 23 days.