Shew that x12-y12 = M (1365), if x and y are prime to 1365. Ex. 5. Ex. 6. Shew that, if m and n are primes; then mn−1 +nm−1 −1=M (mn). Ex. 7. Shew that, if m, n and p are all primes; then (np)m-1+(pm)n−1 + (mn)p− 1 − 1 = M (mnp). Ex. 8. Shew that the 4th power of any number is of the form 5m or 5m+1. Ex. 9. Shew that the 12th power of any number is of the form 13m or 13m+1. Ex. 10. Shew that the 8th power of any number is of the form 17m or 17m ± 1. 378. To find the number of divisors of a given number. Let the given number, N, expressed in prime factors, be a b3c2....... Then it is clear that N is divisible by every term of the continued product (1 + a + a2+...+a*)(1 + b + b2 +...+b3)(1 + c + c2 +...+c2)... Hence the number of divisors of N, including N and 1, is (x + 1) (y + 1) (≈ + 1).............. Ex. 1. The number of divisors of 600, that is of 23.3.52, is (3 + 1) (1 + 1) (2 +1)=24. Ex. 2. Find the sum of the divisors of a given number. The given number being N=a*b*c*..., the sum required is easily seen to be Ex. 3. Find the number of divisors of 1000, 3600 and 14553. Ans. 16, 45, 24. Ex. 4. Shew that 6, 28 and 496 are perfect numbers. [A perfect number is one which is equal to the sum of all its divisors, not considering the number itself as a divisor.] Ex. 5. Find the least number which has 6 divisors. Ans. 12. Ans. 144. Ans. 240. Ex. 7. Find the least number which has 20 divisors. Ex. 8. Find the least numbers by which 4725 must be multiplied in order that the product may be (i) a square, and (ii) a cube. Ans. 21, 245. S. A. 31 379. To find the number of pairs of factors, prime to each other, of a given number. Let the given number be N = a*b*c*...; then, if one of two factors prime to each other contains a, the other does not; and so for all the other different prime factors. Hence the factors in question are the different terms in the product (1+ a*) (1+b) (1+c)..., the number of them being 2", where n is the number of different prime factors of N. The number of different pairs of factors prime to each other is therefore 2"-1, in which result N and 1 are considered as one pair. 380. To find the number of positive integers which are less than a given number and prime to it. Let the given number be N = a*b*c..., where a, b, c,... are the different prime factors of N. The terms of the series 1, 2, 3,..., N which are divisible N by a are a, 2a, 3a...., a; and therefore there are α N a N b numbers which are divisible by a. So also there are We will now shew that every integer which is less than N and not prime to N is counted once and once only in the series Suppose an integer is divisible by only one prime factor of N, a suppose; then that integer is counted once in (a), namely as one of the by a. N α numbers which are divisible Next suppose an integer is divisible by r of the prime factors a, b, c,...; then that integer will be counted r on. Hence the whole number of times an integer divisible by r of the prime factor is counted, is Thus every number not prime to N is counted once in (a); and therefore the number of positive integers less than N and not prime to N is given by (a); provided however that unity is considered to be prime to N. Hence the number of positive integers less than N and prime to N is Ex. 1. Find the number of integers less than 100 and prime to it. Ex. 2. Find the number of integers less than 1575 and prime to it. Ans. 719. Ex. 3. Shew that the number of integers, including unity, which are less than N[N>2] and prime to N is even, and that half N these numbers are less than 2' For if a be prime to N so also is N-a; and if a> N then 381. Forms of square numbers. Some of the different possible and impossible forms of square numbers will be seen from the following examples: Ex. 1. Shew that every square is of the form 3m or 3m+1. For every number is of the form 3m or 3m±1. Hence every square is of the form 9m or 3m+1. Ex. 2. Shew that every square is of the form 5m or 5m±1. For every number is of the form 5m, 5m±1 or 5m±2; and therefore every square is of the form 5m, 5m+1 or 5m+4. Ex. 3. Shew that, if a2 + b2=c2, where a, b, c are integers; then will abc be a multiple of 60. First, every square is of the form 3m or 3m+1; and therefore the sum of two squares neither of which is a multiple of 3 is of the form 3m+2 which cannot be a square. Hence either a or b must be a multiple of 3. Again, every square is of the form 5m or 5m+1. The sum of two squares neither of which is a multiple of 5 is therefore of one of the forms 5m, or 5m ±2. Now no square can be of the form 5m±2; and if a square be of the form 5m, its root must be a multiple of 5. Hence, if ab is not a multiple of 5, c will be a multiple of 5. Thus, in any case, abc is a multiple of 5. Lastly, since every number is of the form 4m, 4m+1, 4m +2 or 4m+3, every square is of the form 16m, 8m+1, 16m+4. Now a and b cannot both be odd, for the sum of their squares would then be of the form 8m+2 which cannot be a square. Also, if one is even and the other odd, the even number must be divisible by 4, for the sum of two squares of the forms 8m+1 and 16m+ 4 respectively is of the form 8m+5 which cannot be a square. It therefore follows that ab must be a multiple of 4. Thus abc is divisible by 3, by 5 and by 4; hence, as 3, 4 and 5 are prime to one another, abc=M (60). Ex. 4. Shew that every cube is of the form 7m or 7m±1. Shew also that every cube is of the form 9m or 9m±1. Ex. 5. Shew that every fourth power is of the form 5m or 5m+1. Ex. 6. Shew that no square number ends with 2, 3, 7 or 8. Ex. 7. Shew that, if a square terminate with an odd digit, the last figure but one will be even. Ex. 8. Shew that the last digit of any number is the same as the last digit of its (4n+1)th power. Ex. 9. Shew that the product of four consecutive numbers cannot be a square. EXAMPLES XXXVIII. 1. SHEW that the difference of the squares of any two prime numbers greater than 3 is divisible by 24. and 2. Shew that, if n be a prime greater than 3, 3. (n + 2m)” − (n + 2m) = M (24). 4. Shew that a1 5. 1m+p — a1n + p = M (30). Shew that, if N − a2 = x and (a + 1)2 – N=y, where x y are positive; then N-xy is a square. 6. How many numbers are there less than 1000 which are not divisible by 2, 3 or 5? 7. P, Q, R, p, q, r are integers, and p, q, rare prime to one another; prove that, if 8. P Ρ Shew that 284 and 220 are two 'amicable' numbers, that is two numbers such that each is equal to the sum of the divisors of the other. 9. Shew that, if 2" - 1 be a prime number, then 2′′-1 (2′′ – 1) will be a 'perfect' number, that is a number which is equal to the sum of its divisors. 10. Find all the integral values of x less than 20 which make x16 - 1 divisible by 680. 16 11. Shew that no number the sum of whose digits is 15 can be either a perfect square or a perfect cube. 12. Shew that every square can be expressed as the difference between two squares. |