V5 - Divisibility Tests, Prime mbe A whole number is a...

V5 - Divisibility Tests, Prime & mbe A whole number is a perfect number if the sum of its factors (other than the number itself) is equal to the number. For example, 6 is a perfect number because its factors sum to 6; that is, $1 + 2 + 3 = 6$. a. Determine which of the following numbers are perfect. i. 28 ii. 40 iii. 14 b. Find at least one more number that is perfect. a. Choose the correct answers below

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.

Sri K.

Question

Please give Ace some feedback