Definitions. 100. A Factor of a number is any one of two or more integers which, multiplied together, produce the number. 101. When one number can be divided by another without remainder, the dividend is said to be divisible by the divisor, and the divisor is called a Measure or Exact Divisor of the dividend. 102. A number that is the product of other factors besides itself and one is called a Composite Number. Note 1.—A Composite number is so called because it is composed of other factors. Note 2. Since a composite number is divisible by its factors, it may be defined to be a number that is divisible by other numbers besides itself and one. 103. A Prime Number is one that has no factors and hence no exact divisor except itself and 1. 104. A Prime Factor is a factor which is a prime number. SLATE EXERCISES. 1. Write in columns the numbers in order from 1 to 35; also from 36 to 70, inclusive, and opposite to each write all the pairs of factors that will produce it. 1=1x1 2=1×2 3=1x3 4=2×2 5=1x5 6=2×3 etc. Thus 36=2×18, 3×12, 4×9, 6x6 38=2 × 19 39=3 x 13 40=2 × 20, 4×10, 5×8 41=1x41 etc. 2. In the same manner write the pairs of factors of numbers from 71 to 107, inclusive; also from 108 to 144. 3. Make a table such as the one required in Ex. 5, p. 53, omitting the first line and first column. Give the factors orally. 4. Make a separate list of numbers from 1 to 144 that have 2 for one or more of their factors. Notice that the right-hand figure of each is or · (?) 5 for one or more of their factors. Notice that the right-hand figure of each is or (?) 3 for one or more of their factors. Divide the sum of the digits of each of these numbers by 3, and notice the remainder, if any. 9 for one or more of their factors. Divide the sum of the digits of each by 9, and notice the remainder, if any. Thus we discover some AIDS IN FINDING FACTORS. 107. It may be shown to be true of any number that it has 2 for a factor if the right-hand figure is 2, 4, 6, 8, or 0; 5 for a factor if the right-hand figure is 0 or 5; 3 for a factor if the sum of its digits is divisible by 3; 9 for a factor if the sum of its digits is divisible by 9. Apply the foregoing aids in the following exercises : 1. Tell which of the dividends at the top of p. 82 are divisible by 2; by 5; by 3 by 9. 2. Write 10 numbers of three or more figures each, all of which shall be divisible by 2; by 5; by 3; by 9. 3. Change one figure in each of the following numbers, so as to make the number divisible by 2: (State what change you make, and why.) 4.-6. Change one figure in each, so that the number shall be divisible by 5.-Change the last figure in each, so that the number shall be divisible by 3.-Change the first figure in each, so that the number shall be divisible by 9. Factoring. 108. Since 7 is a factor of 14..." be a factor of any muun ber of times 14, as 28, 42, etc. That is true al vays that A factor of a factor-of-a-n, aber is itself. Reco" of the number Hence, having obtained one prime factor of a number directly from the number itself, a second one may be obtained from the quotient of the first, and a third, if any, from the quotient of the second, etc. Thus 109. Rule.-Divide the given number by any prime factor, and if the quotient is not a prime number, divide it in like manner, and so continue to divide till the quotient is a prime number. The divisors and the last quotient are the factors sought. Note. The learner will avoid useless labor if he will keep it in mind that the quotient is as much a factor of the dividend as the divisor itself. Suppose, for instance, that he is working to find the prime factors of 479, as in the last of the preceding examples. He tries successively every prime number from 2 upward, till he comes to 23, when he finds that the quotient has become less than the divisor. Here, if he stops to think, he will say to himself: "It is of no use to try any further. This number can not have an exact divisor greater than 23, for if it had, it would have another less than 23; but I have tried every prime number from 2 to 23, and I know it has none. This number is prime." Common Factors and Common Divisors. 110. A factor that occurs in each one of two or more numbers is a Common Factor of those numbers. Thus 15-3 × 5 and 21=3x7; the factor 3 is common to 15 and 21. Note. A factor is said to be common to two or more numbers, just as we may say that the letter i is common to all the syllables of the word Mississippi. III. A factor of a number being an exact divisor of the number, a factor that is common to two or more numbers is a common divisor of those numbers. Find the prime factors of 252 and 2310, and show that the common factors are common divisors of those numbers. 112. The product of any two or more prime factors of a number being an exact divisor of that number, the products of the prime factors common to two or more numbers are common divisors of those numbers. Show that the products of the prime factors that are common to 294 and 315 are common divisors of those numbers. 113. Since an exact divisor of a number can have no factor which does not occur in that number, the product of all the prime factors that are common to two or more numbers is the greatest common divisor of those numbers. Find, if you can, any other divisors of 396 besides its prime factors, and the products of two or more of them. Can 1820 and 2310 have any other common divisors than their common factors and their products? Try to find one. 114. Hence, to find the greatest common divisor of two or more numbers Rule-1. Resolve the given numbers into their prime factors. 2. Multiply together all the factors that are common to all the numbers. The product will be the greatest common divisor sought. Example. Find the g. c. d. of 546 and 910 ? |