still retained in this country to some extent. At the forma. tion of the Constitution, the continental currency had suffered a greater depreciation in some of the colonies than in others. Thus, while a pound in New England was worth $3.33}, in Pennsylvania it was but $2.66, and in New York but $2.50. The value in Federal Money, of the old currencies of the different States, is as follows: A shilling of New England, Virginia, Kentucky, or Tennessee, is 163 cents. A shilling of New York or North Carolina, is 12 cents. A shilling of New Jersey, Pennsylvania, Delaware, or Maryland, is 13} cents. A shilling of South Carolina, or Georgia, is 21% cents. CHAPTER XXII. DIVISIBILITY OF NUMBERS. Every number that cannot be divided by any other number, (except 1,) without a remainder, is called a PRIME NUMBER. Two or more numbers that have no common divisor, are said to be prime to each other. Every prime number is prime to all other numbers except its own multiples. There are no known means of determining at once whether A proposed number is a prime, but the following properties and rules will enable us to determine all the divisors of any number. 1. 2 is a factor of all numbers terminated by 0, 2, 4, 6, or 8. For, as 2 will divide 10, it will also divide any number of tens, or any number of tens plus 2, 4, 6, or 8. Numbers divisible by 2 are called EVEN,_all others, ODD numbers. 2. 5 is a factor of all numbers terminated by 0 or 5. For, as 5 will divide 10, it will also divide any number of tens, or any number of tens plus 5. 3. 3, or 9, is a factor of all numbers in which the sum of the figures is exactly divisible by 3, or 9. For, if from any power of 10, as 10, 100, 1000, &c., we subtract 1, the remainder consists entirely of 9's, and is, therefore, divisible by both 3 and 9. Hence, any power of 10 is divisible by 3 and 9 with 1 remainder, therefore, any number of tens, hundreds, thousands, &c., diminished by as many units, will be divisible by 3 and by 9. Let us, then, examine the number 34794. 3 ten thousands 3 ; 4 thousands - 4; 7 hun. dreds -7; 9 tens — 9; and 4 units — 4; each divided by 3 or 9, give no remainder. Therefore, 34794 3—4—7—9—4, is divisible by 3 and by 9, and if the sum of the numbers sub. tracted, or in other words, the sum of the digits, is similarly divisible, the number itself will be so. 4. 11 is a factor of all numbers in which the sum of the odd digits, (the 1st, 3d, 5th, &c.,) and the sum of the even digits, (the 2d, 4th, 6th, &c.,) are equal, or their difference is some multiple of 11. For any number of tens, thousands, hundred thousands, &c., (which represent the even digits,) increased by as many units, will be divisible by 11. Any number of hundreds, ten thousands, millions, &c., (which re. present the odd digits,) diminished by as many units, will also be divisible by 11. Take, then, the number 635173. 6 hun. dred thousands + 6; 3 ten thousands - 3; 5 thousands + 5; 1 hundred -1; 70 + 7; and 3 3; each divided by 11 give no remainder. Therefore, 635173 18 + 7 or 635173 11, is divisible by 11, and 635173 itself must be so. 5. 4 is a factor of all numbers, in which the two terminating figures are divisible by 4. For, as 4 will divide 100, it will also divide any number of hundreds, or any number of hundreds plus any number of units divisible by 4. 6. 25 is a factor of all numbers terminated by 25, 50, 75, or two zeroes. For, as 25 will divide 100, it will also divide any number of hundreds, or any number of hundreds plus 25, 50, or 75. 7. Every number that is divisible by two or more numpers prime to each other, is divisible by their product. Take for example, 105 which is divisible by both 3 and 5. This be resolved into the factors 5 x 21 ; 5 X 21 must, therefore, be divisible by 3. But as 3 will not divide number may 5, it must divide the other factor 21, and the number may be resolved into the factors 5 X 3 X 7 or 15 X 7. Hence we deduce the following additional properties. 8. Every even number that is divisible by 3 is also divi. sible by 6 ; and every even number that is divisible by 9 is also divisible by 18. 9. Every number divisible by 3 or 9, in which the two terminating figures are divisible by 4, is divisible by 12 or 36. 10. Every number divisible by 3 or 9, whose terminating digit is 0 or 5, is divisible by 15 or 45. 11. Every prime number greater than 2, is one greater or one less than some multiple of 4. 12. Every prime number greater than 3, is one greater or one less than son multiple of 6. 13. Every number that has no prime factor, equal to, or less than its square root, is itself a prime number. For the product of any two factors, each greater than the square root of a number, would evidently be greater than the number itself. Therefore, if we attempt the division of any supposed prime, by all the primes less than its square root, and discover no factor, the number is itself a prime. TO FIND ALL THE DIVISORS OF A NUMBER. What numbers will divide 5940 without a remainder ? 215940 We first resolve the number into all its prime factors, by 2,2970 commencing with 2 and dividing as often as possible, by 31485 each of the prime numbers in succession. We thus find 3 495 that 41580 =22x 33 x 5 x 11, or 2x2x3X3X3X5x11. 3 165 It may, therefore, have as many composite divisors as we 5 55 can form distinct products of these prime factors. In order 11 11 to determine all the possible products, we arrange 1, with 1 the powers of the factor that is employed the greatest num ber of times, in a horizontal line. We then multiply each of the numbers in the first line, by each of the powers of another factor,-each of the numbers of the preceding lines, by each of the powers of a third factor, &c., as in the following table. 45 1 3 9 27 = 33 13 54 = 33 x 2 135 = 33 x 5 10 30 90 270 33 x 2 x 5 20 60 180 540 = 33 x 22 x 5 11 33 99 297 = 33 11 22 66 198 594 = 33 x 2 x 11 44 132 396 1188 38 x 22 x 11 55 165 495 1485 33 x 5 x 11 110 330 990 2970 33 x 2 x 5 x 11 220 660 1980 5940 = 33 x 22 x 5 x 11 The numbers of the first line having been arranged as directed, we multiply them separately by 2 and 22. All the numbers of these three lines, are multiplied by 5, which gives us three new lines of divisors. All the numbers of these six lines, are multiplied by 11, which gives us six new lines of divisors. We thus obtain 48 numbers that will divide 5940 without a remainder, and an examination of the table will show that these are all the divisors, since the prime factors are combined in every pos. sible way. We are able to determine without actual trial, the number of exact divisors of any given number. By the foregoing table we perceive that 33 had 4, or 3 + 1 divisors. 33 x 22 has 12, or 3 + 1 x 2 + 1. 33 x 22 x 5 has 24 or 3 + 1 x 2 + 1 x 1 +- 1. In like manner each new factor can be multiplied by all the preceding divisors, as many times as are equivalent to the exponent of its power, thus forming so many new divisors, to be added to the preceding. Hence, for finding the number of divisors of any given number, we have the following RULE. Add 1 to the exponent of each of the prime factors of the given number, and multiply together the exponents thus increased. The product thus obtained, is the number of divisors sought. 18 * 104 59 2 We have already seen that the greatest common divisor of two or more numbers, may be readily obtained by the aid of a table of prime factors. But by resolving fractions by inspection, into their prime factors, we may often reduce them to their lowest terms, without finding the greatest common divisor. For example, let it be required to reduce 35, 132, 341 68, 105, 279, and 385. to their lowest terms. Resolving each fraction into its prime factors, we have 4 x 17 3 x 5 x 7 3 x 3 x 31 5 x 7 x 11 and 5 X 17 3 X 4 X 11 11 x 31 5 x 11 x 19 Cancelling the factors common to the numerators and denominators, we have , 44. i 19 1 for the lowest terms of each fraction. 1. Resolve 65340 into its prime factors. 2. Find all the divisors of 1200; of 1620. 3. How many integral divisors has 1844 ? 4. How many integral divisors has 1900 ? 5. Reduce 49.15 to its lowest terms. 6. Is 479 a prime number? 7. Is 30907 a prime number? 8. Reduce 1154 to its lowest terms. 9. Reduce 542 to its lowest terms. 10. How many integral divisors has 13600? 11. How many integral divisors has 13475! 12. What are the integral divisors of 700? 13. What are the integral divisors of 1584? 14. What are the integral divisors of 2310? 15. What are the prime factors of 1770! 16. What are the prime factors of 13470 ? 17. How many integral divisors has 95875? 18. Reduce 706- to its lowest terms. 19. Reduce 826 4 to its lowest terms. 20. Reduce 1335 to its lowest terms. 21. Is 21479 a prime number ? 10813 10943 95 |