12. If both dividend and divisor are multiplied by the same number, what will be the effect on the quotient (or fraction) ? If both be divided by the same number, what will be the effect ? [See the third principle developed in this section.] 13. [Write the following exercises on the board, and the answers, as fast as they are given by the class, as follows: 1 of 1=}, &c.] What is 1 of 1? 1 of 2 ? 1 of 3? Of 6 ? Of 7? What is t of 1? Of 2 ? Of 3? Of 4? What is of 1? Of 2? Of 3? Of 4? What is of 1? Of 2? Of 3? Of 4? e of 1? Of 2? Of 3? Of 4 ? Now look at the board, and say, did you get these answers by adding, subtracting, multiplying, or dividing? What is of 1 ? of 1 ? of 1? of? What is the operation, then, when the numbers on each side of of are both fractions ? What, then, does the word of imply, when connected with fractions ? Remember, then, that the word of connected with fractions always implies multiplication. What is the divisor in [? In ? Perform the division in both these cases ? Does the numerator, then, always express the fraction when the denominator is 1 ? Does the value of an integer, then, remain unchanged when 1 is placed under it as a denominator ? 14. What principle is involved in the last exercise ? Ans. A whole number may be expressed fractionally by writing 1 under it as a denominator. 15. Recapitulation. — What effect is produced on a fraction by multiplying its numerator? By multiplying its denominator? By dividing its numerator? By dividing its denominator? By multiplying both its terms? By dividing both its terms ? What does the word of imply when connected with a fraction? How may an integer be expressed fractionally ? 16. What are the principles developed in this section? Ans. 1. If multiplication or division be performed on the numerator, the same effect is produced on the fraction. 2. If multiplication or division be performed on the denominator, a contrary effect is produced on the fraction. 3. No change of value is produced on the fraction when both terms are multiplied or divided by the same number. 4. A whole number may be expressed fractionally by writing 1 under it as denominator. 5. The word of connected with a fraction implies multiplica tion. SECTION II.- Prime Factors, Common Multiples, and Common Divisors. Definitions. - I. Numbers may be divided into two classes, namely, Prime numbers and Composite numbers. A prime number is a number which can be divided exactly only by itself, or by unity, as 1, 2, 3, 5, 7, 11, 17. A composite number is a number which can be measured exactly by a number exceeding unity, or which can be formed by multiplying two or more numbers together, each exceeding unity, as 4 from 2x2; 12 from 2x2x3; 18 from 2X3X3. One number is said to be prime to another when unity is the only integer by which both can be measured. Thus, 4 and 9 are neither of them prime numbers, but they are prime to each other ; because unity is the only integer which will measure them both. II. A number greater than unity that will exactly divide two or more numbers, is called their common divisor ; and the greatest number that will so divide them is called their greatest common divisor. Thus, 5 is a common divisor, and 10 the greatest common divisor of 10 and 50; and 3 is a common divisor, and 9 the greatest common divisor of 9, 18, and 27. III. A number that contains another an exact number of times, is a multiple of that number. Thus 4, 6, and 8, are each multiples of 2. A number that contains two or more numbers as factors is a common multiple of those factors. Thus, 6 is a common multiple of 2 and 3; and 24 is a common multiple of 2, 3, and 4. The smallest number that contains two or more numbers as factors, is their least common multiple. Thus, 24 is a common multiple of 2 and 3; but it is not their least common multiple, for 18 and 12 contain them also; but as no number smaller than 6 contains them, 6 is their least common multiple. 1. What is a prime number ? Give examples, and say why they are prime. What is a composite number? Give examples, &c. What is a common divisor of two or more numbers ? Give examples, &c. What is the greatest common divisor of two or more numbers ? Give examples, &c. What is a mul. tiple of a number? Give examples, &c. What is a common multiple of two or more numbers ? Give examples, &c. What is a least common multiple of two or more numbers ? Give examples, &c. 2. Is 2 a prime or a composite number ? What is an even number ? Can any even number except 2, then, be prime ? Can the product of two or more odd factors ever be even ? Why not? Ans. Because every even number has for one of its factors. From the answers to these questions, it is plain that the most simple method of resolving a number into its prime factors is to continue halving it as long as it remains even (each operation giving 2 for a factor), and then to examine it by the odd numbers, beginning with the smallest. Thus, to resolve 120 into its prime factors, say 2• [60] =2•2 [30] =2•2• 2 [15]=2•2•2• 3.5. These operations, however, should be performed mentally, unless the number be very large; and, after a little practice, the three 2's may be discovered at once (and the same with other numbers), by a mere inspection of the composite number. The following exercises will render the resolution of composite numbers sufficiently easy, and, after some practice, exceedingly rapid. 3. Is 10 divisible by 2 without remainder ? Are 2 tens? 3 tens? Any number of tens? How many tens in 80 ? In 380 ? 270 ? 1250 ? Are all these, numbers, then, divisible by 2 ? Since any number of tens, then, is divisible by 2, by what rank of figures can it be determined whether a number is divisible by 2? What will be the respective remainders, if any, on dividing the following numbers by 2? 154? 379 ? 1976 ? 3285 ? [These numbers, and others, if necessary, to be written on the black-board. They are not to be divided. The pupil should tell at a glance.] What, then, is the sign that 2 is a factor in a number? Ans. That it is an number, or a number ending in 4. What will be the remainder if 10 be divided by 9 ? When 2 tens are divided by 9? 3 tens? 5 tens? 7 tens ? What will remain if 100 be divided by 9 ? 200 ? 500 ? Any number of hundreds ? When 1000 is divided by 9 ? 2000 ? 3000 ? Any number of thousands ? [Write 253674 on the black-board, and other numbers, if necessary, and ask such questions as follow, pointing to the figures.] What will be the remainder when this 2 is divided by 9? The 6? 5? 4? 7 ? 3? What, then, is the sign that 9 is a factor ? Ans. That the sum of the significant figures is divisible, &c. 5. What is the remainder when 1 is divided by 3 ? When 2? 3? 4 ? 5? 6? 7 ? 8? 9 ? When 10 is divided by 3? 2 tens? 3 tens? 4 tens ? &c. When 1000? 2000 ? or 3000 ? &c. What, then, is the sign that 3 is a factor ? That the sum of the significant, &c. 6. [Write the following and other numbers on the blackboard, and ask questions as follows.] 2768, 32562, 5237, 8246. What is the remainder when this number is divided by By 3 ? By 9 ? Note. — All numbers divisible by 9 are also divisible by 3; but the converse is not always true. For instance, 51, 111, 213, are divisible by 3, but not by 9. 7. Is 100 divisible by 4? 200 ? 300 ? Any number of hundreds ? What, then, is the sign that 4 is a factor ? Ans. That the tens and units are, &c. Is 20 divisible by 4 ? 40 ? 60 ? 80 ? Any even number of tens? Give another sign of 4, then. Unite the two signs. Ans. When the units and tens are divisible, if the tens are odd; when the units alone are divisible, if the tens are even. 8. Is 10 divisible by 5? Any number of tens? What. then, is the sign that 5 is a factor? Is 100 divisible by 25 ? What, then, is the sign of 25 ? 9. Is 1000 divisible by 8? What, then, is the sign of 8 ? Is 200 divisible by 8 ? Any even number of hundreds ? Give another sign of 8, then. Unite the two signs. 10. What is the sign of ten being a factor ? Of 50? Of 125=(1900)? Of 225 (9X25) ? 11. If an apple be divided into 2 equal parts, and each of these be again divided into 3 equal parts, into how many equal parts will the apple be divided ? Would the number of parts have been precisely the same, had the apple been divided at once into 6 parts ? Divide 24 by 2, and the quotient by 3. Divide 24 by 6, the product of 2 and 3. Is it the same thing, then, in all cases, whether we divide by two numbers separately, or by their product ? 12. Is 3 a factor in 342 ? Is 2? Is 6 a factor, then? Is 2 a factor in 546 ? Is 3? Is 6 a factor, then? What is the sign of 6 ? An even number divisible by what ? 13. What is the sign of 15 (3X5) ? Of 18 (2X9) ? Of 20 ? Ans. An even number of — What is the sign of 24 (3 X 8) ? Of 75 (3 X 25)? Can an even number be prime? Why? Can a number ending in 5 be prime? Why? 14. [Write 4236981 on the black-board, and frequently exercise the class as follows, on this and other large numbers, till the questions can be answered correctly and rapidly.] What is the remainder, if any, when divided by 2? By 3 ? 4? 5? 9? 25 ? 50 ? 125 ? Is it divisible by 6 ? 15 ? 18? 20 ? 24? 75 ? 225 ? Signs to discover the factors 7 and 11 might be developed, but they are too numerous to be of much use in practice. It may be well to mention, however, that 7 is factor in all numbers consisting of 2 or of 3 figures, where the left hand figure, or figures, is double that of the right hand, as 42, 84, 63, 168, 147, &c.; and also if the right hand figure be of the left hand ones, as 91, 182, 273, 364, &c. Eleven is readily discovered in 2 figures, since they must be alike, as 44, 77; it is also a factor in numbers consisting of 3 figures, when the sum of the figures at the right and left is equal to that in the middle, as 473, 374, 286, 385, &c. [The class may now be directed individually to form on the slate, a table of numbers, from 1 to 100, or to 1000, or to 10,000, as the teacher may see fit, analyzed to their prime factors. When this is done, let the pupils exchange slates, and each examine his school-mate's work, and mark any errors he may find. Lastly, let the tables be carefully examined by the teacher, to see that no composite numbers are placed among the prime factors, which will probably be the case on the first trial, such as 20=4.5, in place of 20=2•2•5. Writing such a table twice without copy or assistance, will generally make every pupil sufficiently familiar and ready with analysis. If not, the exercise should be repeated till he becomes apt in observing the prime factors in a number, and can declare at once how often they are repeated.] Examples of Numbers resolved to Prime Factors. Resolve the following numbers into their prime factors: 1st. 132; 2d. 625; 3d. 488. Solution 1st. 132. Is 2 a factor in 132 ? How often? Is 3? Then 2•2• 3 being factors, 4.3=12. What is the quotient of 132 by 12 ? Is 11 a prime? Then the prime factors of 132 are 2•2•3•11. 2d. 625. Is 2 a factor in 625 ? Is 3 ? Is 5 ? How often ? How many 25s, then, in 625 ? Ans. There being 4 in each hundred, of course there are 25 in 625. Then 25 being twice a factor, gives 5.5.5.5 for the prime factors of 625. 3d. 488. How often is 2 a factor in this number? What is the quotient of 488 by 8 ? Is 61 prime or composite? Then the prime factors of 488 are |