Factors and Coprimes 2/1

1. Find the product of all the factors of 50
a. $${50}$$
b. $${50^2}$$
c. $${50^3}$$
d. $${50^4}$$

2. P is the product of all the factors of 15552. If P = ${12^N} \times M$, where M is not a multiple of 12, then find the value of M. [M and N are positive Integers]
a. ${3^{42}}$
b. ${3^{46}}$
c. ${3^{48}}$
d. ${3^{52}}$

3. Let M be the set of all the distinct factors of the number N=${6^5} \times {5^2} \times 10$,Which are perfect squares. Find the product of the elements contained in the set M.
a. $${5^{20}}$$
b. $${5^{22}}$$
c. $${5^{24}}$$
d. $${5^{26}}$$

4. In a hostel there are 1000 students in 1000 rooms. One day the hostel warden asked the student living in room 1 to close all the doors of the 1000 rooms. Then he asked the person living in room 2 to go to the rooms which are multiples of his room number 2 and open them. After he ordered the 3rd student to reverse the condition of the doors which are multiples of his room number 3. If He ordered all the 1000 students like the same, Finally how many doors of those 1000 rooms are in open condition?
a. 31
b. 168
c. 169
d. 969

5. What is the product of all factors of the number N = ${6^4} \times {10^2}$ which are divisible by 5?
a. ${2^{210}} \times {3^{140}} \times {5^{105}}$
b. ${2^{206}} \times {3^{130}} \times {5^{105}}$
c. ${2^{225}} \times {3^{180}} \times {5^{125}}$
d. ${2^{215}} \times {3^{140}} \times {5^{125}}$

6. Let N = ${2^3} \times {3^{17}} \times {5^6} \times {7^4}$ and M = ${2^{12}} \times {3^5} \times {5^4} \times {7^8}$. P is total number of even factors of N such that they are not factors of M. Q is the total number of even factors of M such that they are not factors of N. Then 2P -Q = ?
a. 40
b. 126
c. 69
d. 195