12 [Hint: of the in are of n, so tha22. Given n 1, let o,(n) the sum of the sth of the divisors ofsuffices to p*, p is a prime.]111NUMBER-THEORETIC FUNCTIONSCiven a positive integer k 1, show that there are infinitely many integers n for whichO = k, but at most finitely many n with o (n) = k.a Show that there are no positive integers n satisfying o (n) = 10.IHint: Note that for n 1, o (n) n.]a Prove that there are infinitely many pairs of integers m and n with o (m2) = o (n²).Hint: Choose k such that gcd(k, 10) = 1 and consider the integers m = 5k, n = 4k.]%3D%3D%3Da) n= 2k-1 satisfies the equation o (n) = 2n – 1.b) If 2* – 1 is prime, then n = 2*='(2* – 1) satisfies the equation ơ (n) = 2n.e) If 2k – 3 is prime, then n = 2k-'(2k - 3) satisfies o (n) = 2n + 2.It is not known if there are any positive integers n for which o (n) = 2n + 1.15. If n and n + 2 are a pair of twin primes, establish that o (n +2) = o (n) +2; this alsoholds for n = 434 and 8575.16. (a) For any integer n 1, prove that there exist integers n1 and n2 for which%3D|— и (в)-%3DyT HI (ɔ)%3Dt(n1) + t(n2) = n.(b) Prove that the Goldbach conjecture implies that for each even integer 2n there existintegers n¡ and n2 with o (n1) + o (n2) = 2n.17. For a fixed integer k, show that the function f defined by f(n) = nk is multiplicative.18. Let f and g be multiplicative functions that are not identically zero and have the propertythat f(p*) = g(p*) for each prime p and k 1. Prove that f = g.19. Prove that if f and g are multiplicative functions, then so is their product fg and quotienf/g (whenever the latter function is defined). (0-)3020. Let w(n) denote the number of distinct prime divisors of n 1, with w(1) = 0. Foinstance, w(360) = w(2³ . 3² . 5) = 3.(a) Show that 2@(n) is a multiplicative function.(b) For a positive integer n, establish the formula%3Dwound%3D%3D7(n²) = 20(d)%3Du|pPor any positive integer n, prove that Eain T(d)' = (Ed
T(d)).mini: Both sides of the equation in question are multiplicative functions of n, so thasuffices to consider the case n =%3Dven n 1, let o,(n) denote the sum of the sth powers of the positive divisors ofthat is,sP

Question

12 [Hint: of the in are of n, so tha22. Given n > 1, let o,(n) the sum of the sth of the divisors ofsuffices to p*, p is a prime.]111NUMBER-THEORETIC FUNCTIONSCiven a positive integer k > 1, show that there are infinitely many integers n for whichO = k, but at most finitely many n with o (n) = k.a Show that there are no positive integers n satisfying o (n) = 10.IHint: Note that for n > 1, o (n) > n.]a Prove that there are infinitely many pairs of integers m and n with o (m2) = o (n²).Hint: Choose k such that gcd(k, 10) = 1 and consider the integers m = 5k, n = 4k.]%3D%3D%3Da) n= 2k-1 satisfies the equation o (n) = 2n – 1.b) If 2* – 1 is prime, then n = 2*='(2* – 1) satisfies the equation ơ (n) = 2n.e) If 2k – 3 is prime, then n = 2k-'(2k - 3) satisfies o (n) = 2n + 2.It is not known if there are any positive integers n for which o (n) = 2n + 1.15. If n and n + 2 are a pair of twin primes, establish that o (n +2) = o (n) +2; this alsoholds for n = 434 and 8575.16. (a) For any integer n > 1, prove that there exist integers n1 and n2 for which%3D|— и (в)-%3DyT HI (ɔ)%3Dt(n1) + t(n2) = n.(b) Prove that the Goldbach conjecture implies that for each even integer 2n there existintegers n¡ and n2 with o (n1) + o (n2) = 2n.17. For a fixed integer k, show that the function f defined by f(n) = nk is multiplicative.18. Let f and g be multiplicative functions that are not identically zero and have the propertythat f(p*) = g(p*) for each prime p and k > 1. Prove that f = g.19. Prove that if f and g are multiplicative functions, then so is their product fg and quotienf/g (whenever the latter function is defined). (0-)3020. Let w(n) denote the number of distinct prime divisors of n > 1, with w(1) = 0. Foinstance, w(360) = w(2³ . 3² . 5) = 3.(a) Show that 2@(n) is a multiplicative function.(b) For a positive integer n, establish the formula%3Dwound%3D%3D7(n²) = 20(d)%3Du|pPor any positive integer n, prove that Eain T(d)' = (Ed
 T(d)).mini: Both sides of the equation in question are multiplicative functions of n, so thasuffices to consider the case n =%3Dven n> 1, let o,(n) denote the sum of the sth powers of the positive divisors ofthat is,sP

Accepted Answer

(a)
To find all positive integer n such that τn=10.
τn is the total number of positive divisor of n.…

Find the form of all positive integers n satisfying t(n) = 10. What is the smallest positive integer for which this is true? khow that there are no positive integers n satisfying o (n) = 10. %3D [Hint: Note that for n > 1, o (n) > n.] ve that there are infinitely many pairs of int