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

View Answer

Explanation: The prime factorization of 1800 is 19 * 2

^{2}* 5

^{2}and

S(2

^{2}) = 1 + 2 + 4 = 7

S(5

^{2}) = 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)

View Answer

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

View Answer

Explanation: The prime factorization of 1800 is 3 * 2

^{7}* 5

^{2}and

S(3) = 1 + 3 = 4

S(2

^{2}) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255

S(5

^{2}) = 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

View Answer

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

View Answer

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

View Answer

Explanation: The prime factorization of 1872 is 13 * 3

^{2}* 2

^{4}and S(2

^{4}) = 1 + 2 + 4 + 8 + 16 = 31, S(5

^{2}) = 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

View Answer

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

View Answer

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

View Answer

Explanation: The prime factorization of 1800 is 431 * 2

^{2}* 5 and

S(2

^{2}) = 1 + 2 + 4 = 7

S(5

^{2}) = 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

View Answer

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)(3 + 1)(1 + 1) = 6 * 4 * 2 = 48 odd positive factors.

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