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Problem 1. Find the prime factorization of the following numbers: (a) 269. (b) 311. (c) 721. (d) 1001. (e) 20!. (f) gcd(20!, 2^8 × 3^5 × 7^10 × 334). (g) lcm(20!, 2^8 × 3^5 × 7^10 × 334). (h) gcd(p^a × q^b × r^c, p^d × q^e × r^f), where p, q, r are distinct positive primes and a, b, c, d, e, f are non-negative integers. (i) lcm(p^a × q^b × r^c, p^d × q^e × r^f), where p, q, r are distinct positive primes and a, b, c, d, e, f are non-negative integers. Problem 2. Find the total number of positive divisors of the following numbers: (a) 2^3 × 3^5. (b) 2^10 × 3^20. (c) p^a × q^b, where p, q are distinct positive primes and a, b are non-negative integers. (d) p1^a1 × · · · × pn^an, where p1, . . . , pn are distinct positive primes and a1, . . . , an are nonnegative integers. (e) 669. Problem 3. Find the sum of all positive divisors of 669. Problem 5. Show that a^2 | b^2 if and only if a|b. Hint: use prime factorization for the hard implication.

          Problem 1. Find the prime factorization of the following numbers: (a) 269. (b) 311. (c) 721. (d) 1001. (e) 20!. (f) gcd(20!, 2^8 × 3^5 × 7^10 × 334). (g) lcm(20!, 2^8 × 3^5 × 7^10 × 334). (h) gcd(p^a × q^b × r^c, p^d × q^e × r^f), where p, q, r are distinct positive primes and a, b, c, d, e, f are non-negative integers. (i) lcm(p^a × q^b × r^c, p^d × q^e × r^f), where p, q, r are distinct positive primes and a, b, c, d, e, f are non-negative integers.

Problem 2. Find the total number of positive divisors of the following numbers: (a) 2^3 × 3^5. (b) 2^10 × 3^20. (c) p^a × q^b, where p, q are distinct positive primes and a, b are non-negative integers. (d) p1^a1 × · · · × pn^an, where p1, . . . , pn are distinct positive primes and a1, . . . , an are nonnegative integers. (e) 669.

Problem 3. Find the sum of all positive divisors of 669.

Problem 5. Show that a^2 | b^2 if and only if a|b. Hint: use prime factorization for the hard implication.

Added by Stephen S.

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