SECTION IX. PROPERTIES OF NUMBERS, TESTS OF DIVISI. BILITY, FACTORS, MULTIPLES, DIVISORS. 101. Definitions. (a.) A FACTOR of any given number is such a number as taken an entire number of times will produce the given number; or, the FACTORS of a number are the numbers which multiplied together will produce it. Thus, 2 is a factor of 4, 6, 8, &c.; 3 is a factor of 6, 9, 12, 15, &c. (6.) A DIVISOR of a number is any number which will exactly divide it. NOTE. — Every divisor of a number must be a factor of it, and every factor of a number a divisor of it. The terms factor and divisor, as here used, are nly applied to entire numbers. (..) A PRIME NUMBER is one which has no other factors besides itself and unity. Thus, 1, 2, 3, 5, 7, 11, 13, 17, &c., are prime numbers. (d.) A COMPOSITE NUMBER is one which has other factors besides itself and unity. Thus, 4, 6, 8, 9, 10, 12, 14, 15, 16, &c., are composite numbers. (e.) Any entire number of times a given number is a MULTIPLE of it; or, a MULTIPLE of a number is any number which can be exactly divided by it. Thus, 12 is a multiple of 1, 2, 3, 4, 6, and 12, because it is an exact number of times each of them, or because it can be divided by each without a remainder. (f.) Two numbers are PRIME TO EACH OTHER when they have no common factor. For example, 4 and 9 are prime to each, as are 8 and 15, 24 and 35, &c. Again, 6 and 9 are not prime to each other, because they have the common factor 3; 8 and 12 are not prime to each other, because they have the common factor 4, &c. NOTE. — It is obvious from the foregoing, that every number is a factor of all its multiples, and a multiple of all its factors. 102. Demonstration of Principles. Proposition First. — If one of two numbers is a factor of another, it must be a factor of any number of times that other number. For to find any number of times a given number, we have only to multiply the number by some new factor, without striking out any of the former ones. Illustrations. Since 2 is a factor of 12, it must be a factor of any number of times 12, as 24, 36, 48, &c. Since 7 is a factor of 14, it must be a factor of any number of times 14, as 28, 42, 56, &c. Proposition Second. - If each of two numbers is a multiple of a third number, their sum and their difference must also be multiples of that third number. For, adding an exact number of times a given number to, or subtracting it from, an exact number of times the same number, must give an exact number of times that number. Illustrations. 8 times 9, or 72, + 3 times 9, or 27, =ll times 9, or 99. So 8 times 9, or 72, 3 times 9, or 27, = 5 times 9, or 45. Again. Both 12 and 20 are multiples of 4, and so is their sum, 32, and their difference, 8. Both 42 and 28 are multiples of 7, and so is their sum, 70, and their difference, 14. Proposition Third. - If one of two numbers is a multiple of a third number, and the other is not, neither their sum nor their difference will be a multiple of that third number. For both the sum and the difference of an entire, and a fractional, number of times a given number, must equal a fractional number of times that given number. Illustrations. 8 times 6, or 48, + 24 times 6, or 15, = 103 times 6, or 63. 8 times 6, or 48, — 24 times 6, or 15, = 54 times 6, or 33. 74 times 9, or 66, 4 times 9, or 36, = 37 times 9, or 30. 74 times 9, or 66, + 4 times 9, or 36, =lig times 9, or 102. Again. 20 is a multiple of 5, and 13 is not; hence, neither their sum, 33, nor their difference, 7, is a multiple of it. 38 is not a multiple of 6, and 24 is; hence, neither their sum, 62, nor their difference, 14, is a multiple of it. Proposition Fourth. If neither of two numbers is a multiple of a third, their sum or their difference may or may not be a multiple of it. The truth of this proposition can best be made manifest by a few illustrations. 1. Neither 7 nor 23 is a multiple of 2; yet both their sum, 30, and their difference, 16, are multiples of 2. 2. Neither ; nor 28 is a multiple of 3; yet their sum, 33, is, and their difference, 23, is not, a multiple of 3. 3. Neither 8 nor 17 is a multiple of 3; yet their sum, 25, is not, and their difference, 9, is, a multiple of 3. 4. Neither 14 nor 27 is a multiple of 4; and neither their sum, 41, nor their difference, 13, is a multiple of 4. 103. Tests of the Divisibility of Numbers. Application of the foregoing propositions. 1. Divisibility by 2, 5, 31, or by any other number which will exactly divide 10. Every number greater than ten is composed of a certain number of tens, plus the units expressed by its right hand figure. But the part which is made up of tens must (102, Prop. I.) be divisible by any divisor of ten; and hence, (102, Prop. II. and III.) the divisibility of the entire number will depend on the part expressed by the right hand figure. Therefore, a number is divisible by 2, 5, 23, 14, or by any other number which will exactly divide 10, when its right hand figure is thus divisible. Illustration. 4125 is divisible by 5, by 21, and by 1, because each of these numbers will exactly divide 10, and also 5, the right hand figure of the given number. II. Divisibility by 4, 20, 25, 50, 124, 164, or any other number which will exactly divide 100. Every number greater than 100 is composed of a certain number of hundreds, plus the number expressed by its two right hand figures. But the part which is made up of hundreds must (102, Prop. I.) be divisible by any divisor of one hundred, and hence, (102, Prop. II. and III.) the divisibility of the entire number must depend on the part expressed by the two right hand figures. Therefore, a number is divisible by 4, 20, 25, 50, 121, or by any other number which will exactly divide 100, where its two right hand figures are thus divisible. III. Divisibility by 8, 40, 125, 250, 500, 333, 1663, or by any other number which will exactly divide 1000. Every number greater than 1000 is composed of a certain number of thousands, plus the number expressed by its three right hand figures. But the part which is composed of thousands must (102, Prop. I.) be divisible by any divisor of 1000, and hence, (102, Prop. II. and III.) the divisibility of the entire number must depend on the three right hand figures. Therefore a number is divisible by 8, 125, 250, 500, 1663, 3331, or by any number which will exactly divide 1000, when its three right hand figures are thus divisible. NOTE. Similar tests for determining the divisibility of numbers by any divisor of 10,000, 100,000, &c., could be established, but as they could very rarely be applied to advantage, we omit them. IV. Divisibility by 9. (a.) If 1 be subtracted from a unit of any decimal denomination above unity, the remainder will be expressed entirely by O's, and will therefore be a multiple of 9. Illustrations. 10--1=9 1= 99 1000 1= 999 10000 -l=9999 &c., &c. (6.) But unity, or 1,=0X9+1; and if, for convenience of statement, we regard 0 X 9, or 0, as a multiple of 9, it 100 |