# Discrete Mathematics Questions and Answers – Counting – Number and Sum of Divisors

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This set of Advanced Discrete Mathematics Questions and Answers focuses on “Counting – Number and Sum of Divisors”.

1. Calculate sum of divisors of n = 1900.
a) 6530
b) 5346
c) 3387
d) 4123

Explanation: The prime factorization of 1800 is 19 * 22 * 52 and
S(22) = 1 + 2 + 4 = 7
S(52) = 1 + 5 + 25 = 31
Therefore, S(1800) = 19 * 7 * 31 = 4123.

2. Given the factorization of a number n, then the sum of divisors can be computed in _______
a) linear time
b) polynomial time
c) O(logn)
d) o(n+1)

Explanation: The exact number of running time depends on the computational model. When analyzing arithmetic with large numbers, we usually count either bit operations or arithmetic operations of size O(logn) (where n is the input size). Now, given the factorization of a number n, then the sum of divisors can be computed in polynomial time.

3. Calculate the sum of divisors of N = 9600.
a) 23250
b) 47780
c) 54298
d) 31620

Explanation: The prime factorization of 1800 is 3 * 27 * 52 and
S(3) = 1 + 3 = 4
S(22) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255
S(52) = 1 + 5 + 25 = 31
Therefore, S(1800) = 4 * 255 * 31 = 31620.

4. Find the number of odd positive integers of the number 456.
a) 54
b) 27
c) 16
d) 8

Explanation: To find the number of odd factors (which includes 1), we can exclude any power of 2 and do the same. So, for 456, we have (3 + 1)(1 + 1) = 8 odd positive factors.

5. The number of even positive integers of 3200 is _______
a) 24
b) 32
c) 164
d) 209

Explanation: To find the number of even factors, we can multiply the number of even factors by the power of 2. For 3200, we have (5 + 1)(1 + 1)(2) = 24 even factors.

6. What is the sum of divisors of the number 1872?
a) 12493
b) 5438
c) 45862
d) 654

Explanation: The prime factorization of 1872 is 13 * 32 * 24 and S(24) = 1 + 2 + 4 + 8 + 16 = 31, S(52) = 1 + 5 + 25 = 31. Therefore, S(1872) = 31 * 31 * 13 = 12493.

7. Find the odd positive integer of the number 6500.
a) 43
b) 17
c) 12
d) 87

Explanation: To find the number of odd factors, we can exclude any power of 2 and do the same. So, for 6500, we have (5 + 1)(1 + 1) = 6 * 2 = 12 odd positive factors.

8. How many even positive integers are there in the number 7362?
a) 16
b) 58
c) 35
d) 165

Explanation: To find the number of even factors, we can multiply the number of even factors by the power of 2. For 5065, we have (3 + 1)(1 + 1)(2) = 4 * 2 * 2 = 16 even factors.

9. Calculate sum of divisors of n = 8620.
a) 7549
b) 54201
c) 18102
d) 654

Explanation: The prime factorization of 1800 is 431 * 22 * 5 and
S(22) = 1 + 2 + 4 = 7
S(52) = 1 + 5 = 6
Therefore, S(1800) = 6 * 7 * 431 = 18102.

10. Find the odd positive integer of the number 4380.
a) 108
b) 48
c) 75
d) 8 