The result obtained by multiplying a number by an integer is called a multiple of the number.

Thus the multiples of 8 are 8, 16, 24, 32, 40, 48, etc. The Least Common Multiple of two or more numbers is the least number that is a multiple of each of the numbers. In other words, the Least Common Multiple of two or more numbers is the least number that is exactly divisible by each of the numbers.

Least Common Multiple is denoted by L. C. M.

Example 1. What is the L. C. M. of 8 and 12?
Writing multiples of 8 we have many, including:
8 16 24 32 40 48 64 72 80

and of 12,

12 24 36 48 60 72 84

Notice that 24 is a common multiple of 8 and 12. So also are 48 and 72. 24 is the smallest or least common multiple (L. C. M.).

Example 2. What is the L. C. M. of 12 and 18?

Make a table of multiples of 12 and 18. Then 36, or 2 times 18, is the L. C. M. of 12 and 18.

Example 3. A man buys two kinds of sugar, one kind in 4-pound bags, and the other in 5-pound bags. What is the smallest number of pounds of each kind he can buy so as to have the same amount of each?

Here the answer is the L. C. M. of 4 and 5. The L. C. M. of 4 and 5 is 20. Hence, he buys 20 pounds of each kind.

Since the L. C. M. of two or more numbers is exactly divisible by each of the numbers, it follows that the L. C. M. contains all the factors of each of the given numbers.

This fact suggests a method of finding the L. C. M. of two or more numbers.

Example: Find the L. C. M. of 48, 60, 72.

48 24 × 3. The prime factors are 2 and 3.

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60 = 22 × 3 × 5. The prime factors are 2 and 3 and 5. 72 = 28 × 32. The prime factors are 2 and 3. Any multiple of 48 must contain 2 four times as a factor. Any multiple of 60 must contain 5 as a factor.

Any multiple of 72 must contain 3 twice as a factor.

Hence, the number 24 x 32 × 5 = 720 contains all the factors of the three numbers 48, 60, 72. Therefore, the L. C. M. of 48, 60, 72, is 720.

To find the L. C. M. of two or more numbers, resolve each of the numbers into its prime factors, then find the product of all the prime factors of the given numbers, taking each factor the greatest number of times it occurs in any of the numbers.

Another Method

Example: Find the L. C. M. of 48, 60, 72, 90.

248 60 72 90

224 30 36 45

2 12 15 18 45 L.C.M=15x2x3x2x2x2=720. 3 6

2

9 45
15

NOTE. As 15 in the third horizontal row is a factor of 45, it may be canceled out; also 9 in the following row.

Step 1. Arrange the numbers in a horizontal row.

Step 2. Divide by a prime factor common to two or more of the numbers. Set down the quotients and the undivided numbers.

Step 3. Treat the second horizontal row in the same manner, and so on until a horizontal row is obtained that consists only of numbers prime to one another of which no two are alike if larger than 1.

The continued product of the numbers in the last row and of the divisors will be the L. C. M.

EXPLANATION.

This method is really the same as the other, in principle; each divisor used representing one of the factors which must appear in the L. C. M., while the quotients must include the other factors of each number. Thus: in the given example, the first division shows that the L. C. M. of 48, 60, 72, and 90 must contain the factor 2, and the L. C. M. of 24, 30, 36, and 45. But the next division shows that the L. C. M. of 24, 30, 36, and 45 must contain the factor 2, and the L. C. M. of 12, 15, 18, and 45. Similarly, the L. C. M. of 12, 18, and 45 must contain the factor 2, and the L. C. M. of 6, 9, and 45. But the L. C. M. of 6 and 45 must contain the factor 3, and the L. C. M. of 2 and 15. The L. C. M. of 2 and 15 is 2 x 15 or 30. Therefore the L. C. M. of 48, 60, and 72 is 2 x 2 x 2 x 3 x 2 x 15 or 720.

Review carefully the definitions and explanations of fractions, given on pages 5 and 6.

What is a fraction? How is a fraction written? What does the numerator express? What does the denominator express? What is a proper fraction? An improper fraction? A mixed fraction? (See pages 5 and 6.)

A fraction may also be considered as an indicated division. The numerator is the dividend and the denominator is the divisor.

Is of $1 equal to † of $2?
Is of $1 equal to of $3?
Is of $1 equal to } of $4?

Is of 1 foot equal to † of 3 feet?
Is of 1 gallon equal to of 7 gallons?

Is of 1 yard equal to 1 of 2 yards?

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REDUCTION OF FRACTIONS

Is 14 feet of a foot? Is 1=1?

Is of an hour = 4 hours? Is 2 = 41?
Is of an apple of an apple? Is

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?? Changing the form of a fraction without changing its value is called reduction of fractions.

An improper fraction is reduced to a mixed number by performing the division of numerator by denominator and expressing the result as a whole or mixed number.

Thus, 17 reduces to 41.