Proof. We have mentioned that the 12. and (b u|p u|p 13. P the stated result follows immediately from Theorem 6.4. 14. I PROBLEMS 6.1 Let m and n be positive integers and p1, P2, least one of m or n. Then m and n may be written in the form .. k1 k2 m = Pj P2 with k; > 0 for i = 1, 2, . , r 15. %3D .. n = p{' p.…· p ji „j2 with j; 2 0 for i = 1, 2, . , r 16 ... Prove that lcm(m, n) = p{ P2 P d..dd = (u'u)pɔ where u; = min {k¡, ji}, the smaller of k; and ji; and v; = max {k;, ji}, the larger oft. and ji. 1. 1. Use the result of Problem 1 to calculate gcd(12378, 3054) and lcm(12378, 3054). Deduce from Problem 1 that gcd(m, n) lcm(m, n) In the notation of Problem 1, show that gcd(m, n) = 1 if and only if kiji = 0 for i = 1, 2, ... , r. (a) Verify that t(n) = t(n + 1) = t(n +2) = t(n + 3) holds for n = 3655 and 4503. (b) When n = 14, 206, and 957, show that o (n) = o (n + 1). 6. For any integer n > 1, establish the inequality t(n) < 2./n. [Hint: If d | n, then one of d or n/d is less than or equal to An.1 7. Prove the following. (a))t(n) is an odd integer if and only if n is a perfect square. O g(n) is an odd integer if and only if n is a perfect square or twice a perfect square. = mn for positive integers m and n. %3D %3D %3D %3D %3D %D [Hint: If p is an odd prime, then 1+ p + p²+ 8) Show that an 1/d = o (n)/n for every positive integer n. If n is a square-free integer, prove that t (n) = 2, where r is the number of prime divisors ...+ ) + p* is odd only when k is even. of n. 10. Establish the assertions below: (a) If n = p p.. p is the prime factorization of n > 1. then (u)) (b) For any positive integer n, P2 1- >1+ 23 (c) If n > 1 is a composite number, then o (n) [Hint: See Problem 8.] >n+n.