Least Common Multiple.

200. Multiples. If a number is multiplied by an integer, the product is called a multiple of the number.

Thus, $20 is a multiple of $5, since 4 times $5 is $20.

201. A series of multiples of a number is found by multiplying the number by the integers, 1, 2, 3, 4, 5, etc. Since a composite number is the product of only one set of prime numbers (§ 174), every multiple of a number contains all the prime factors of the number.

202. Common Multiple. A multiple of two or more numbers is called a common multiple of the numbers.

Thus, 6 x $2 = $12; 4 × $3 = $12; 3 × $4 = $12; 2 × $6 = $12. Therefore, $12 is a common multiple of $2, $3, $4, and $6.

203. Least Common Multiple. The smallest common multiple of two or more numbers is called their least common multiple; and it is the smallest number that is exactly divisible by each of them.

Thus, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, etc.; and the multiples of 4 are 4, 8, 12, 16, 20, 24, etc. The common multiples of 3 and 4 are 12, 24, etc.; and the smallest of these is 12. Therefore, the least common multiple of 3 and 4 is 12.

204. The letters L. C. M. stand for the words Least Common Multiple.

205. The L. C. M. of two or more numbers is a number that contains all the prime factors of each of these numbers. Every prime factor, therefore, must occur in the L. C. M. the greatest number of times it occurs as a factor in any one of them.

Thus, $20

=

2 × 2 × 5 × $1, and $30 = 2 × 3 × 5 × $1. The L. C. M. of $20 and $30 is, therefore, 2 × 2 × 3 × 5 × $1, or $60.

206. Find the L. C. M. of 84, 168, 252, and 420.

SOLUTION. Resolve each of the numbers into its prime factors.

The factor 2 occurs three times in 168; the factor 3 occurs twice in 252; the factor 5 occurs once in 420; and the factor 7 occurs once in all the given numbers.

Therefore, the L. C. M. is 28 × 32 × 5 × 7= 2520. Hence,

207. To Find the L. C. M. of Two or More Numbers, Separate each number into its prime factors. Find the product of these factors, taking each factor the greatest number of times it occurs in any one of the given numbers.

208. Examples. 1. Find the L. C. M. of 18, 24, 27, 45. Arrange the numbers in line, and divide by the smallest prime number that will divide two or more of the numbers.

We first divide by 2, and write the quotients and undivided numbers in a line below. In the first line of quotients we cancel 9, as it is an exact divisor of 27, and, therefore, 27 contains all the factors of 9. We next divide by 3, and the quotients by 3, and obtain the numbers 4, 3, 5 in the last line. No two of the numbers 4, 3, and 5 have a common factor. Hence, the L. C. M. is 2 × 3 × 3 × 4 × 3 × 5 = 1080.

2. Find the L. C. M. of 3, 9, 27, 54.

3, 9, 27, 54.

We cancel the 3, which is contained in 9; then the 9, which is contained in 27; then the 27, which is contained in 54, and have 54 for the L. C. M. of the numbers.

8. Find the L. C. M. of 13, 15, 26, 39:

We cancel the 13 of the first line and divide by 3, getting 5, 26, 13. We cancel the 13 of this line. The L. C. M. is 3 × 5 × 26 390.

209. If the given numbers are large and contain no prime factors that can readily be detected, it is best to obtain the common factors by the process for finding the

G. C. M. under like circumstances.

Example. Find the L. C. M. of 1247 and 1769.

1247)1769(1

1247

2 522

9 261

29)1247(43

116

87

87

Hence, the G. C. M. of 1247 and 1769 is 29; and 124729 × 43,

and 1769 29 × 61.

Therefore, the L. C. M. of 1247 and 1769 is 29 × 43 × 61 = 1247 × 61, or 1769 × 43, that is, 76,067.

210. From this process it will be seen that :

The L. C. M. of two numbers may be found by dividing either of the numbers by their G. C. M. and multiplying the quotient by the other number.

211. The L. C. M. of two prime numbers, or of two numbers prime to each other, is their product.

CHAPTER VII.

COMMON FRACTIONS.

212. What is the name of one of the parts when a unit

214. When a unit is divided into twelve equal parts,

215. Equal parts of a unit are called fractional parts of the unit.

216. In three quarters of a yard, the unit counted is a quarter of a yard.