16. What is the greatest common divisor of 16, 24, and 36?

In solving Ex. 16, we first find the divisor of 16 and 24, viz. 8, and then find the divisor of 8 and 36; or first find the divisor of 24 and 36, viz. 12, and then of 12 and 16; or we might first find the divisor of 16 and 36, and then of that divisor and 24.

17. What is the greatest common divisor of 84, 96, 144, and 174?

18. What is the greatest common divisor of 77, 105, and 140? 19. What is the greatest common divisor of 9 and 16?

Ans. 1. 20. What is the greatest common divisor of 9, 12, and 20?

LEAST COMMON MULTIPLE.

124. A MULTIPLE of a number is any number which is divisible by that number; thus, 15 is a multiple of 5 and also of 3; 21 is a multiple of 7 and of 3.

NOTE. Every number is both a divisor and a multiple of itself.

125. A COMMON MULTIPLE of two or more numbers, is any number which is divisible by each of the given numbers; thus, 48 is a common multiple of 4, 6, and 8.

123.

How is Ex. 16 solved ? 124. What is a Multiple of a number! 125. A Common Multiple of two or more numbers?

126. The LEAST COMMON MULTIPLE of two or more numbers, is the least number that is divisible by each of the given numbers; thus, 24 is the least common multiple of 4, 6, and 8.

NOTE. There is no such thing as a least common divisor, or greatest common multiple.

127. PROBLEM 3. To find the least common multiple of two or more numbers.

Ex. 1. What is the least common multiple of 20, 24, and 36? Ans. 2 X2 X 2 × 3 × 3 × 5 = 360.

no number less than 360 will contain 20, 24, and 36, for if one of the 2's in the common multiple were omitted, it would not contain 24; if one of the 3's, it would not contain 36; and if the 5 were omitted, it would not contain 20.

Similar reasoning applies in all examples.

Hence,

RULE 1. Resolve each number into its prime factors, and the continued product of all the different prime factors, each taken the greatest number of times it occurs in either of the given numbers, will be the least common multiple.

2. What is the least common multiple of 12, 16, 20, and 30? Ans. 2 X 2 X 2 × 2 × 3 × 5 240.

3. What is the least common multiple of 22, 33, and 55 ? 4. What is the least common multiple of 16, 36, 40, and 48? 5. What is the least common multiple of 20, 30, 50, and 80? 6. What is the least common multiple of 15, 25, 45, and 75? 7. What is the least common multiple of 35, 50, 75, and 90? 8. What is the least common multiple of 24, 36, 48, and 64? 9. What is the least common multiple of 72, 80, 84, and 96? 10. What is the least common multiple of 42, 49, 72, and 88?

126. The Least Common Multiple? May numbers have a least common divi visor? Greatest common multiple? 127. Rule for finding the least common multiple? Reason?

128. The same result is sometimes more easily attained by RULE 2. Having set the given numbers in a line, divide by any PRIME number that will divide two or more of them, and set the quotients and undivided numbers in a line beneath; proceed with this line as with the first, and so continue until no two of the numbers can be divided by any number greater than one; the continued product of the divisors and numbers in the last line will be the rultiple sought.

The second rule may be illustrated by the example already employed in explaining the first rule, viz.:

What is the least common multiple of 20, 24, and 36? Ans. 2 X2 X3 X5 X 2 X 3360.

OPERATION.

2) 20, 24, 36

2) 10, 12, 18
3) 5, 6, 9

If the process by the 1st rule be examined it will be seen that the factor 2 is found 7 times in the given numbers, and as 2 is taken but 3 times in finding the multiple, it is rejected 4 times. By the 2d rule, also, 2 is rejected 4 times, viz. twice in the 1st division by 2 and twice in the 2d division by 2. The learner may think 2 is rejected three times in each of the two first divisions, but he must remember that the divisor, 2, is retained as a factor in the common multiple in each instance.

5, 2, 3

Similar remarks are applicable to all rejected factors in like examples, .. the two rules give the same result.

il. What is the least common multiple of 5, 16, 24, 32, and 48? Ans. 2 X2 X2 X2 X2 X3 X5

By Rule 1.

5=5

OPERATION.

By Rule 2.

480.

2) 5, 16, 24, 32, 48

NOTE 1. The principle, which is the same in the two rules, is most readily perceived by the first operation.

12. What is the least common multiple of 30, 40, 45, and 75? 13. What is the smallest sum of money with which I can buy horses at $50 each, cows at $30 each, or sheep at $8 each, using the same sum in each case? Ans. $600.

14. I have 4 wine measures; the first holds 4 quarts, the seoond 5 quarts, the third 6 quarts, and the fourth 8 quarts; what is the size of the smallest cask that can be exactly measured by means of each of these measures? Ans. 120 quarts. 15. What is the least common multiple of 10, 15, 45, 75, and

90?

In solving Ex. 15, it is evident that 10, 15, and 45 may at once be struck out, for each of these numbers is a measure of 90, and .. whatever multiple of 75, and 90 is found, it, certainly, must be a multiple of 10, 15, and 45; hence, the question is reduced to this: What is the least common multiple of 75 and 90?

NOTE 2. Many other abbreviations of this and other rules may be effected, but a delicate perception of the relations of numbers, and a skillful application of principles, will much more facilitate the progress of the learner than any set of formal rules.

(a) If the numbers are prime, or even mutually prime, their product is their least common multiple.

16. What is the least common multiple of 9 and 10?

Ans. 9 × 10

17. What is the least common multiple of 8, 9, and 25 ?

90.

(b) The least common multiple of two numbers is equal to their product divided by their greatest common divisor.

18. What is the least common multiple of 12 and 20?
The greatest common divisor of 12 and 20 is 4, and
The least common multiple is 12 X 20 4: 60, Ans.
19. What is the least common multiple of 63 and 72?
20. What is the least common multiple of 33 and 77 ?

128. Ex. 15, how solved? What of other abbreviations? ultiple of mutually prime numbers? Of two numbers?

Least common

COMMON FRACTIONS.

129. A FRACTION is an expression representing one o more of the equal parts of a unit.

NOTE. A unit, or any other whole number, is often called an Inteyer, It is also called an Integral or Entire Number.

130. A COMMON or VULGAR FRACTION is expressed by two numbers, one above and the other below a line; thus (one half), (two fifths), etc.

(a) The number below the line shows into how many parts the unit is divided, and is called the DENOMINATOR, because it denominates or gives name to the parts; thus, if a unit is divided into 3 equal parts, each part is one third; if into 8, each part is one eighth; etc?

(b) The number above the line is called the Numerator, because it numerates or numbers the parts taken.

(c) The numerator and the denominator are the TERMS of the fraction.

131. A fraction is nothing more nor less than unexcuted division, i. e. division indicated but not performed, the numerator being the dividend, and the denominator the divisor. Hence,

(a) The value of a fraction is the quotient of the numerator, divided by the denominator; thus, 4212÷4 = 3; and, ...,

(b) Any change in the NUMERATOR causes a LIKE change in the value of the fraction, and any change in the DENOMINATOR causes an OPPOSITE change in the value of the fraction (Art. 84).

These principles are developed in the following Problems.

129 What is a Fraction? Other names for a whole number? 130. A Com. mon Fraction, how expressed? Number below the line, what called? Why? Number above, what called? Why? Terms of a fraction, what! 131. A frac tion, what is it? Value of a fraction? What follows?