Question

Define a function S:Z + →Z + as follows. For each positive integer n,S(n)= the sum of the positive divisors of n. Find the following. (For each answer, enter an exact number.)

(a) S(1)=
(b) S(15)=
(c) S(13)=
(d) S(7)=
(e) s(18)=
(f) S(21)=

Answer 1

The function S(n) provides the sum of all positive divisors of a positive integer n. For the given numbers, S(1) is 1, S(15) is 24, S(13) is 14, S(7) is 8, S(18) is 39, and S(21) is 32.

Step-by-step explanation:

To find S(n), where S is defined as the function that gives the sum of all positive divisors of a positive integer n, we first need to list out all divisors of n and then sum them up. Let's calculate following the defined function S:

• S(1): The only divisor of 1 is 1 itself. Hence, S(1) = 1.
• S(15): The divisors of 15 are 1, 3, 5, and 15. Therefore, S(15) = 1 + 3 + 5 + 15 = 24.
• S(13): Since 13 is a prime number, its only divisors are 1 and 13. Thus, S(13) = 1 + 13 = 14.
• S(7): 7 is also a prime number, so its divisors are 1 and 7. This implies that S(7) = 1 + 7 = 8.
• S(18): The divisors of 18 are 1, 2, 3, 6, 9, and 18. So, S(18) = 1 + 2 + 3 + 6 + 9 + 18 = 39.
• S(21): The divisors of 21 are 1, 3, 7, and 21. Therefore, S(21) = 1 + 3 + 7 + 21 = 32.