PROPERTIES OF PRIME NUMBERS. 171. No direct process of detecting prime numbers has been discovered. NOTE. A few facts, such as are given below, if kept in mind, will aid somewhat in ascertaining whether a number is prime or not. 172. The only even prime number is 2; since all other even numbers, as 4, 6, 8, and 10, it is evident, can be exactly divided by 2, and therefore must be composite. 173. The only prime number having 5 for a unit or righthand figure is 5; since every other whole number thus terminating, as 15, 25, 35, and 45, can be exactly divided by 5, and therefore must be composite. 174. Every prime number, except 2 and 5, must have 1, 3, 7, or 9 for the right-hand figure; since all other numbers are composite. 175. Every prime number above 3, when divided by 6, must leave 1 or 5 for a remainder; since every prime number above 3 is either 1 greater or 1 less than 6, or some exact number of times 6. 176. In a series of odd numbers written in their proper or natural order, if beginning with 3 every THIRD number, with 5 every FIFTH, with 7 every SEVENTH, be cancelled, as composite, the remaining numbers, with 2, will be the prime numbers of the natural series. Thus, in the series 1, 3, 5, 7, 9, 11, 13, X5, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, every third number from the 3, every fifth from the 5, every seventh from the 7, every ninth from the 9, and so on, being cancelled, the remaining numbers, with 2, are all the prime numbers under 50. NOTE 1. In the series, every third number from the 3 contains that number as a factor; every fifth number from the 5, that number as a factor; and so on. NOTE 2. The whole number of prime numbers from 1 to 100,000 is 9,593. Although all of these, except 2 and 5, end in 1, 3, 7, or 9, there are, within the same range, no less than 30,409 composite numbers terminating with some one of the same figures. 177. All the prime numbers not larger than 4057 are in 191 491 829 2351 193 499 839 197 503 853 199 509 857 211 521 859 223 523 863 227 541 877 229 547 881 1193 1567 1973 2741 3187 3583 4007 3593 4013 FACTORING. 178. FACTORING is the process of resolving a quantity into its factors. 179. Every number that is not prime is composed of prime factors, since all numbers are either prime or composite; and, if composite, can be separated into factors, which, if themselves composite, can be further separated into those that shall be prime. 180. To resolve a composite number into its prime factors. Ex. 1. It is required to find the prime factors of 42. OPERATION. 2142 321 Ans. 2, 3, 7. The We divide by 2, the least prime number greater than 1, and obtain the quotient 21; and, since 21 is a composite number, we divide this by 3, and obtain for a quotient 7, which is a prime number. several divisors and the last quotient, all being prime, constitute all the prime factors of 42, which, multiplied together, they equal. Hence 7 Divide the given number by any prime number that will exactly divide it, and the quotient, if a composite number, in the same manner; and so continue dividing, until a prime number is obtained for a quotient. The several divisors and the last quotient will be the prime factors required. NOTE 1.- The composite factors of any number may be found by multiplying together two or more of its prime factors. NOTE 2. Such prime factors as two or more numbers may have alike, are termed prime factors common to them; and these may be readily determined after the numbers are resolved into their prime factors. EXAMPLES. 2. What are the prime factors of 105? 3. Resolve 220 into its prime factors. 4. What are the prime factors of 936? Ans. 3, 5, 7. Ans. 2, 2, 2, 3, 3, 13. Ans. 2, 3, 31, 67. 5. What are the prime factors of 1953 ? 6. Resolve 12462 into its prime factors. 7. Resolve 19987 into its prime factors. 8. What are the prime factors common to 225, 435, and 540 ? Ans. 3, 5. 9. What are the prime factors common to 960, 1568, and 5824? 10. What are the prime factors common to 2340, 11934, 12987, and 14859? Ans. 3, 3, 13. 11. A man has 105 apples, which he wishes to distribute into small parcels, each of equal numbers; what are the smallest whole numbers, greater than 1, into which they may be exactly divided? Ans. 3, 5, and 7. DIVISIBILITY OF NUMBERS. Thus, 181. One number is said to be divisible by another, when the latter will divide the former without a remainder. 9 is divisible by 3. 182. One number is divisible by another, when it contains all the prime factors of that number. Thus, 12, which contains all the factors of 4, is divisible by 4. 183. All even numbers, or such as terminate with 0, 2, 4, 6, or 8, are divisible by 2, since each of them contains 2 as a facThus, 10, 24, 36, 58, are each divisible by 2. tor. 184. All numbers which terminate with 0 or 5 are divisible by 5, since each of them contains 5 as a factor. Thus, 20, 25, 50, are each divisible by 5. 185. Every number is divisible by 4, or any other number that will exactly divide 100, when its two right-hand figures are divisible by the same. For any figure on the left of the two right-hand figures must express one or more hundreds, and a factor of one hundred is a factor of any number of hundreds; so, if the sum exactly divides the units and tens of a number, the entire number will be divisible by it. Thus, 116 is divisible by 4; 140, by 20; 225, by 25; and 450, by 50. 186. Every number is divisible by 8, or any other number that will exactly divide 1000, when its three right-hand figures are divisible by the same. For any figure on the left of the three right-hand figures must express one or more thousands, and a factor of one thousand is a factor of any number of thousands; so, if the sum exactly divides the units, tens, and hundreds of a number, the entire number will be divisible by it. Thus, 1824 is divisible by 8; 1840, by 40; 3375, by 125; 2750, by 250; and 4500, by 500. 187. Every number the sum of whose digits 3 or 9 will exactly divide, is divisible by 3 or 9. For 10, or any power of 10, less 1, gives a number, as 9, 99, 999, &c., which is divisible by 3 and by 9. Hence, any number of tens, hundreds, thousands, &c., less as many units, must be divisible by 3 and by 9; and if the excess of units denoted by the significant figures, in the aggregate, is likewise divisible by 3 and by 9, it follows that the entire number is thus divisible. For example, 7542 is a number, the sum of whose digits is divisible by 3 and by 9; and separated into tens, hundreds, and thousands, it is equal to 7000+500 + 40+ 2. Now, 7000 = 7 X 1000 = 7 × (999+1) 7 × 9997; 500 5 × 995; and 40 = = = = 5 X 100 = 4 X 10 4 5 × (99+1) X (91) = 4 X 94. Therefore, 75427 × 999 + 5 × 99 + 4 × 9 + 7 + 5 + 4 + 2. The remainders 7 + 5 + 4 + 2, corresponding with the significant figures of the number, added together, equal 18, which sum being divisible by 3 and by 9, it is evident that 7542 is divisible in like manner. NOTE. Upon the property of 9 now explained depends the method of proving, by excess of nines, multiplication (Art. 63), and division (Art. 75). The same method of proof may be resorted to in addition and in subtraction. Thus, To prove Addition. Find the excess of nines in each number added, and then the excess of nines in the sum of these results; which, if the work be right, will equal the excess of nines in the answer. To prove Subtraction. Find the excess of nines in the subtrahend, and also in the remainder, and then the excess of nines in the sum of these results; which, if the work be right, will equal the excess of nines in the minuend. 188. Every number occupying four places, in which two like significant figures have two ciphers between them, is divisible by 7, 11, and 13. Thus, 9099, 1001, 3003, 4004, &c., are each divisible by 7, 11, and 13. |