6. If the sum of an odd square number and an even square number is also a square number, then the even square number is divisible by 16. 7. Every square number is of the form 5n or 5n ± 1. 8. Every cube number is of the form 7n or 7n± 1. 9. If a number be both a square and a cube it is of the form 7n or 7n + 1. 10. No square number is of the form 3n-1. 11. No triangular number is of the form 3n – 1. 12. If n be any number whatever, a the difference between n and the next number greater than n which is a square number, and b the difference between n and the next number less than n which is a square number, then n ab is a square number. 13. If the difference of two numbers which are prime to each other, be an odd number, any power of their sum is prime to every power of their difference. 14. If there be three numbers one of which is the sum of the other two, twice the sum of their fourth powers is a square number. 15. Shew when n is any prime number, that x2 1 and (x-1)" will leave the same remainder when divided by n. 16. If 2p+1 be a prime number and the numbers 1, 2,...p3, be divided by 2p+ 1, the remainders are all different. 17. Every even power of every odd number is of the form Sn + 1. 18. Every odd power of 7 is of the form 8n-1. 19. If n be any integer, n2 - n+1 cannot be a square number. 20. If n be any odd integer, then n3 + 1 cannot be a square number. 21. If a and x are integers, the greatest value of ax- -2x is the integer equal to or next less than a2 22. Shew that n (n + 1) (2n + 1) is always divisible by 6. 23. If n be odd, (n - 1)n (n + 1) is divisible by 24. 24. by 6. If n be odd and not divisible by 3, then n2 + 5 is divisible 25. If n be a prime number greater than 5, then n1- 1 is divisible by 240. 26. Shew that m5 та m is an integer if m be. 27. Shew that n-n is always divisible by 42. 28. If n be any prime number and a prime to n, prove that x and x when divided by n will leave the same remainder. 29. If n be any prime number and N prime to n, then N3- - 1 is divisible by n3. 30. If n be any prime number greater than 3 and N prime to n, then N" - N is divisible by 6n. 31. If n and N be different prime numbers, and each greater than 3, then N"--1 is divisible by 24n. 32. If n be any prime number greater than 2, except 7, then no 1 is divisible by 56, 33. If n be any prime number greater than 2 and N any odd number prime to n, then N1-1 is divisible by 8n. 34. If n be any prime number greater than 2, then is a multiple of n. 1" + 2" + 3" + +(rn)" 35. Shew that the 10th power of any number is of the form 11n or 11n + 1. 36. Shew that the 12th power of any number is of the form 13n or 13n+ 1. t 37. Shew that the 9th power of any number is of the form 19n or 19n± 1. 38. Shew that the 11th power of any number is of the form 23n or 23n± 1. 39. Shew that the 20th power of any number is of the form 25n or 25n+ 1. 40. How many positive integers are less than 140 and prime to 140? 41. How many positive integers are less than 360 and prime to 360 ? How many positive integers are less than 1000 and prime to 1000 ? 43. How many positive integers are less than 3 x 7a × 11 and prime to it? 44. How many positive integers are less than 10" and prime to it? 46. Find the number of divisors of 1845. 47. Find how many divisors there are of 9, and the sum of these divisors. 48. Into how many pairs of factors prime to each other can 1845 be resolved? 49. In how many ways can a line of 100800 inches long be divided into equal parts, each some multiple of an inch? 50. In how many ways can four right angles be divided into equal parts so that each part may be a multiple of the angular unit, (1) when the unit is a degree, (2) when the unit is a grade? 51. How many different positive integral solutions are there of xy=10" ? 52. If N be any number, n the number of its divisors, and P the product of its divisors, shew that P=N"; shew that N* is in all cases a complete square. 53. Find the least number which has 30 divisors. 54. Find the least number which has 64 divisors of which three are primes whose continued product is 30. 55. Suppose a prime to b, and let the quantities be divided by b; prove that the sum of the quotients arising from any two terms equidistant from the beginning and end will be a 1, and that the sum of the corresponding remainders will be b. 56. If any number of square numbers be divided by a given n number n there cannot be more than different remainders. 2 57. Express generally the rational values of x and y which satisfy 140x=y3. 58. If r, the radix of a scale of notation, be a prime number r+1 greater than 2, there are different digits in which square 2 numbers terminate in that scale. 59. If any number n can be resolved into the sum of p squares, 2 (p-1) n can be resolved into the sum of p (p-1) squares. 60. If n be any positive integer 22+15n −1 is divisible by 9. 61. If P, denote the sum of the products of the first n numbers taken r together, 1 + P+P2+ ... + P, is a multiple of [n. 1 62. Shew that the 100th power of any number is of the form 125n or 125n + 1. LIII. PROBABILITY. 714. If an event may happen in a ways and fail in b and all these ways are equally likely to occur, the probability ways, and the probability of its failing is b a+b' This may be regarded as a definition of the meaning of the word probability in mathematical works. The following explanation is sometimes added for the sake of shewing the consistency of the definition with ordinary language. The probability of the happening of the event must, from the nature of the case, be to the probability of its failing as a to b; therefore the probability of its happening is to the sum of the probabilities of its happening and failing as a to a + b. But the event must either happen or fail, hence the sum of the probabilities of its happening and failing is certainty. Therefore the probability of its happening is to certainty as a to a + b. So if we represent certainty by unity, the probability of the happening of the event is repre 715. Hence if p be the probability of the happening of an event, 1-p is the probability of its failing. 716. The word chance is often used in mathematical works as synonymous with probability. 717. When the probability of the happening of an event is to the probability of its failing as a to b, the fact is expressed in popular language thus; the odds are a to b for the event, or b to a against the event. 718. Suppose there to be any number of events A, B, C, &c., such that one must happen and only one can happen; and suppose a, b, c, &c., to be the numbers of ways in which these events can respectively happen, and that all these ways are equally likely to occur, then the probabilities of the events are proportional to a, b, c, &c. respectively. For simplicity let us consider three events, then A can happen in a ways out of a+b+c ways and fail in b+c ways; therefore, by Art. 714, the probability of and the probability of A's failing is A's happening is a a+b+c |