154. Oral Exercises Using the tests described above, a. Name the numbers expressed in B, page 58, that contain the factor 2; 4; 5. b. Name the numbers in C, page 58, that contain the factor 3; 6; 9. c. Name the numbers in D, page 58, that contain the factor 8; 9; 10; 100. To find the Prime Factors of a Number. 155. ILLUSTRATIVE EXAMPLE I. What are the prime factors of 2205 ? Explanation. - Applying the tests (Art. 153) to WRITTEN WORK. the given number, we find that 2 is not, but that 3 | 2205 3 is, a factor of 2205 ; and, by dividing, see that 3 735 2205 = 3 x 735. 5 245 Seeking, in the same way, a prime factor of 7 49 735, we find that 735 = 3 x 245. Continuing 7 this process, we find that 245 = 5 x 49, and that 49 = 7 x 7. Therefore, 2205 = 3 x 3 x 5 x 7 x 7, and Ans. 3, 3, 5, 7, 7. the prime factors are 3, 3, 5, 7, and 7. 156. ILLUSTRATIVE EXAMPLE II. What are the prime factors of 409 ? Ecplanation. — Applying the tests (Art. 153), we find that 409 19) 409 (21 23) 409 (17 is not divisible by 2, 3, or 5. We 38 23 then try to divide by the other 29 179 prime numbers in order until we 19 161 reach 23, when we see that the quotient is less than the divisor. 10 18 There can then be no prime factor in 409 greater than 23, for if there were, there would be another factor (the quotient) less than 23, which we should have found before reaching 23. The number 409 is therefore prime. 157. As we have found in Art. 155 that 2205 equals the product of all its prime factors, so we shall always find that A composite number equals the product of all its prime factors. WRITTEN WORK. 158. When a composite number is expressed as a product of prime factors, it is said to be separated into its prime factors. 159. From the above examples may be derived the following Rule. To separate a number into its prime factors : 1. Divide the given number by one of its prime factors. 2. Divide the quotient , thus obtained by one of its prime factors, and so continue dividing until a quotient is obtained that is a prime number. 3. This quotient and the several divisors are the prime factors sought. Proof. The product ought to equal the given number. NOTE. If no prime factor is readily found by which to divide, we try to divide by the several prime numbers in order. If no prime factor is found before the quotient becomes less than the trial divisor, the given number is prime. See Illustrative Example II. 160. Examples for the Slate. Separate into prime factors the following numbers: (1.) 180. (4.) 208. (7.) 329. (10.) 644. (2.) 192. (5.) 260. (8.) 338. (11.) 684. (3.) 176. (6.) 169. (9.) 357. (12.) 2500. Select the prime numbers and find the prime factors of the composite numbers among the following: (13.) 341. (18.) 450. (23.) 704. (28.) 945. (14.) 344. (19.) 590. (24.) 711. (29.) 972. (15.) 362. (20.) 560. (25.) 762. (16.) 367. (30.) 2688. (21.) 596. (26.) 808. (31.) 1164. (17.) 408. (22.) 689. (27.) 836. (32.) 3248. SYMBOLS OF OPERATION. 161. The signs +, -, *, and +, since they indicate that certain operations (adding, subtracting, multiplying, and dividing) are to be performed, are called symbols of operation. 162. In expressing a series of operations by aid of these signs, it is often necessary to indicate that an operation is to be performed on two or more numbers combined. This is done by writing the numbers to be operated upon, with the proper signs, and enclosing the whole expression in marks of parenthesis or brackets. The expression so enclosed is then treated as if it denoted a single number. Thus, (7 + 2) × 5 means that the sum of 7 and 2 is to be multiplied by 5; but 7 + 2 x 5 means that 7 is to be increased by 5 times 2. (7 – 2) 3 means that the difference between 7 and 2 is to be multiplied by 3; but 7 - 2 x 3 means 7 diminished by 3 times 2. 7 + 2 (7 + 2) -5, or means that the sum of 7 and 2 is to be 5 divided by 5. [(2 + 3) 5 -11] 2 means that the sum of 2 and 3 is to be multiplied by 5, the product diminished by 11, and the remainder multiplied by 2. 163. In performing a series of operations indicated by signs, First, operate on the numbers that are written within pcrentheses as indicated by the signs. Next, multiply and divide as indicated by the signs x and -. Finally, add and subtract as indicated by the signs + and * The horizontal line here drawn between 7+2 and 5 is equivalent to marks of parenthesis. 164. Oral Exercises. [The Key contains answers to the following examples.) a. (6 + 8) 5 = ? h. 3 84 * 3 ? b. +8 * 5 = ? i. 3 x 8 =(4 x 3) = ? c. (8-3) x 2 = ? x4-2-3 ? d. 8*3 x 2 = ? 2 e. 8 + 12 =4 = ? 8 + 3 8 -3 k. f. (8 +12) - 4 = ? 2 2 g (2+1) (7-2) = ? 1. [(4+6) ~ 4-5 x 3] x 3 = ? j. 14 + CANCELLATION. WRITTEN WORK. Ans. 4. 165. ILLUSTRATIVE EXAMPLE I. If 4 be multiplied by 3 and the product divided by 3, what is the result ? From this example we see that 4 x 3 If a given number be multiplied by a 4 3 number, and the product be divided by the same number, the result will be the given number. In such cases, both the multiplication and the division may be omitted. NOTE. This omission is indicated in the written work above by drawing a mark through the 3 thus, %. 166. ILLUSTRATIVE EXAMPLE II. What is the result of dividing the product of 4 and 6 by 3 ? Explanation. — As 6 2x3, the dividend in this example is 4 x 2 x3, and the divisor is 3, so that we 2 4x6 may strike out the factor 3 in both dividend and 8 divisor, and multiply by 2 only, thus shortening the work. The process of shortening work by striking out equal factors in dividend and divisor is cancellation. WRITTEN WORK. 167. Examples for the Slate. All operations upon numbers should first be indicated, as fur us possible, by signs, that the work to be done may be shortened, if possible, by cancellation. 33. Divide 81 x 42 by 99 x 7. 34. Multiply 75 x 10 by 3 * 6, and divide that product by 15 x 25 x 12. 35. Divide 7 x8 x 48 by 63 x 4 x 5 x 17, and multiply the quotient by 51. 36. If 5 sets of chairs, 6 in a set, cost $ 75, what did 1 chair cost? 37. If it requires 13 bushels of wheat to make 3 barrels of flour, how many bushels will be required to make 78 barrels of flour? 38. If a tree 54 feet high casts a shadow of 90 feet, wha? length of shadow will be cast by a flag-staff 105 feet high? 39. A grocer exchanged 561 pounds of sugar, at 12 cents per pound, for eggs at 22 cents per dozen. How many dozen were received ? 40. If 12 pieces of cloth, each piece containing 62 yards, cost $372, what do 24 yards cost ? 41. If the work of 7 men is equal to the work of 9 boys, how many men's work will equal the work of 90 boys? 42. If 15 men consume a barrel of flour in 6 weeks, how long would it last 9 meu ? 43. If 12 men can build a wall in 42 days, how many days will be required for 21 men to build it? 44. If $ 15 purchase 12 yards of cloth, how many yards will $ 48 purchase? 45. A ship has provision for 15 men 12 months. How long will it last 45 men ? 46. How many overcoats, each containing # yards, can be made from 10 bales of cloth, 12 pieces each. 42 yards in each piece ? |