Rule. Multiply one power of the factor by another until we have a product which contains the factor the required number of times. 2. Find the sixth power of 2; of 5; of 6. Ans. 64; 15625; 46656. 3. Find the twelfth power of 2; of 3; of 4. Ans. 4096; 531441; 16777216. 4. Form a composite number of four equal factors; of five; of six; of seven. 5. Find the 2d, 3d, 5th, 7th, and 10th powers of 3; of 4; of 5. Ans. 59049; 1048576; 9765625, etc. CASE III. 152. To form a composite number out of factors bearing certain relations to each other. 1. Find a composite number of three factors, the smallest being 4, the second being twice, and the third three times the first. SOLUTION.-The number will contain 4 used three times as a factor, and also 2 and 3; hence we raise 4 to the third power, which is 64, and multiply the result by the product of 2 and 3, or 6, which gives 384. OPERATION. 4364 64×6=384 Rule.-Raise the given factor with which the others are compared, to the power indicated, and multiply the result by the product of the factors indicating the relation. NOTE. We may also find each factor and then multiply them together. 2. Form a number of three factors, the first being 3, the second twice, and the third 3 times the first. Ans. 162. 3. Find a number whose smallest factor is 5, the second being twice, and the third 6 times as great. Ans. 1500. 4. Required a number, one of whose factors is 8, another one-half as large, and the third twice as large. Ans. 512. 5. Find a number, two of whose factors are 6, two onehalf as large, and two 3 times as large. Ans. 104976. 6. Find a number one of whose factors is 8, another 4 more, another half the sum of these two, and another half the difference of the first and second. Ans. 1920. CASE IV. 153. To form all the composite numbers possible out of given factors. 1. Form all the composite numbers possible out of 2, 3, and 5. 1-2 OPERATION. 1-3 1-2-3-6 1-5 SOLUTION. It will be seen that if we write 1 and 2 with a hyphen between them, and multiply each term by 1 and 3, which may be written in the same way, we shall have all the numbers which can be obtained from the multiplication of these factors; and if we multiply the series by 1 and 5, which may be written as before, we shall have 1, 2, 3, and 5, and all the composite numbers that can be formed from 2, 3, and 5. Omitting 1, 2, 3, and 5, in the last result, we shall have all the composite numbers that can be formed with 2, 3, and 5. 1-2-3-6-5-10-15-30 Ans. 6, 10, 15, 30 Rule.-I. Connect 1 and the first given factor by a hyphen, and multiply each term by 1 and by the second given factor, and this series by 1 and by the third factor. II. Proceed in the same way till all the factors are used, and the terms of the last product, omitting one and the given factors, will be the numbers required. Form the possible composite numbers 2. Out of 2, 3, 5, and 11. Ans. 6, 10, 15, 30, 22, etc. Ans. 6, 10, 15, 14, etc. Ans. 6, 14, 21, 42, 22, 33, 77, 26, 39, etc. 5. Out of 2, 3, 5, 7, 11, and 13. Ans. 6, 10, 15, 14, 21, 35, 77, 91, 143, etc. CASE V. 154. To form all the composite numbers possible when some of the factors are alike. 1. Find the composite numbers which can be formed out of 2, 2, 2, 3, and 3. SOLUTION. Since 2 is used three times, the first series will evidently be 1-2-22-23, or 1-2-4-8; and since 3 is used twice, the second series will be 1-3-32, or 1-3-9, and the products of these terms, omitting the unit and the factors themselves, will be the composite numbers required. OPERATION. 1-2-4-8 1-2-4-8-3-6-12 Rule.-I. Write 1 and the successive powers of a factor repeated (the highest power being indicated by the number of times the factor occurs) in a row; under this write 1 and the successive powers of another factor, and find the products of the terms of the two series. II. Proceed in a similar manner with the products and the remaining factors; the terms of the last product, omilting 1 and the original factors, will be the numbers required. Form the possible composite numbers out 2. Of 2, 2, 3, 3, and 3. 3. Of two 2's, two 3's, and 5. 4. Of three 2's, two 3's, and 7. 5. Of four 3's, 5, and 7. Ans. 4, 12, 36, 108, etc. Ans. 4, 6, 9, 12, etc. Ans. 4, 8, 24, 72, etc. Ans. 9, 27, 81, 405, etc. 6. Of four 2's, three 3's, two 5's, and 7. Ans. 4, 8, 16, 9, 27, 210, etc. CASE VI. 155. To find the number of composite numbers that can be formed from given factors. 1. How many composite numbers can be formed with three 2's and two 3's? SOLUTION.-2 used three times as a factor gives, with unity, a series of four terms, and three used twice gives a series of three terms; hence their product will give a series of 4×3, or 12 terms; and, omitting the unit and 2 and 3, we will have 9 terms. Hence there will be 9 composite numbers. OPERATION. 1-2-4-8-4 terms. 1-3-93 terms. 4x3=12 Rule. Increase the number of times each factor is used by 1, take the product of the results, and diminish this by the number of different factors used plus one. How many composite numbers can be formed 2. Out of two 3's and two 5's? 4. Out of three 2's, four 3's, and three 5's? Ans. 6. Ans. 17. Ans. 76. Ans. 114. Ans. 66. Ans. 1433 DIVISIBILITY OF COMPOSITE NUMBERS. 156. Composite Numbers can be divided by the factors which produce them. 157. The Factors of many composite numbers may be seen by inspection from the following principles: PRINCIPLES. 1. A number is divisible by 2 when the right hand term is zero or an even digit. For, the number is evidently an even number, and all even numbers are divisible by 2. 2. A number is divisible by 3 when the sum of the digits is divisible by 3. This may be shown by trying several numbers, and, seeing that it is true with these, we infer that it is true with all. A rigid demonstration is given in the latter part of the book. 3. A number is divisible by 4 when the two right hand terms are ciphers, or when the number they express is divisible by 4. If the two right hand terms are ciphers, the number equals a number of hundreds, and since 100 is divisible by 4, any number of hundreds is divisible by 4. If the number expressed by the two right hand digits is divisible by 4, the number will consist of a number of hundreds plus the number expressed by the two right hand digits (thus 1232=1200+32); and since both of these are divisible by 4, their sum, which is the number itself, is divisible by 4. 4. A number is divisible by 5 when its right hand term is 0 or 5. When the unit figure is 0 the last partial dividend must be 0, 10, 20, 30, or 40, each of which is divisible by 5. When the unit figure is 5, the last partial dividend must be 15, 25, 35, or 45, each of which is divisible by 5. 5. A number is divisible by 6 when it is even, and the sum of the digits is divisible by 3. Since the number is even it is divisible by 2, and since the sum of the digits is divisible by 3 the number is divisible by 3, and since it contains both 2 and 3, it will contain their product, 3×2, or 6. 6. A number is divisible by 8 when the three right hand terms are ciphers, or when the number expressed by them is divisible by 8. If the three right hand terms are ciphers, the number equals a number of thousands, and since 1000 is divisible by 8, any number of thousands is divisible by 8. If the number expressed by the three right hand digits is divisible by 8, the entire number will consist of a number of thousands plus the number expressed by the three right hand digits (thus 17368= =17000+368); and since both of these parts are divisible by 8, their sum, which is the number itself, is divisible by 8. 7. A number is divisible by 9 when the sum of the digits is divisible by 9. This may be shown by trying several numbers, and, seeing that it is true with these, we can infer that it is true with ́all. rigidly demonstrated. It may also be 8. A number is divisible by 10 when the unit figure is 0. For, such a number equals a number of tens, and any number of tens is divisible by 10, hence the number is divisible by 10. NOTE.-1. A number is divisible by 7 when the sum of the odd numerical periods, minus the sum of the even numerical periods, is divisible by 7. 2. A number is divisible by 11 when the difference between the sums of the digits in the odd places and in the even places is divisible by 11, or when this difference is 0. 3. These two principles are rather curious than useful. For their demonstration see the latter part of the book, where will also be found quite a full treatment of the Properties of Numbers. FACTORING. 158. Factoring is the process of finding the factors of composite numbers. 159. The Factors of a composite number are the numbers which, when multiplied together, will produce it. Unity and the number itself are not regarded as factors. 160. The Prime Factors of a composite number are the prime numbers which, multiplied together, will produce it. 161. A Root of a number is ONE of its several equal factors; thus 3 is a root of 9, and 4 of 64. 162. The Second Root of a number is one of its two equal factors; thus, since 2×2=4, 2 is the 2d root of 4. 163. The Third Root of a number is one of its three equal factors; the 4th root is one of its 4 equal factors, etc. 164. The Symbol of roots is the radical sign ✔; a small figure placed at the left of the sign, called the index, |