There are exactly ten stores and no other buildings on a straight street. On the northern side of the street, from West to East, are stores 1, 3, 5, 7 and 9; on the southern side of the street, also from West to east, are stores 2, 4, 6, 8 and 10. The stores on the northern side are located directly across the street from those on the southern side, facing each other in pairs, as follows: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10. Each store is decorated with lights in exactly one of the following colours: green, red, and yellow. The stores have been decorated with lights according to the following conditions:
No store is decorated with lights of the same colour as those of any store adjacent to it.
No store is decorated with lights of the same colour as those of the store directly cross the street from it.
Yellow lights decorate exactly one store on each side of the street.
Red lights decorate store 4.
Yellow lights decorate store 5.

Which one of the following could be an accurate list of the colours of the lights that decorate stores 2, 4, 6, 8 and 10, respectively?
A. green, red, green, red, green
B. green, red, yellow, red, green
C. green, red, green, yellow, red
D. yellow, green, red, green, red
E. yellow, red, green, red, yellow

If green lights decorate store 7, then each of the following statements could be false EXCEPT:
A. Green lights decorate store 2
B. Green lights decorate store 10
C. Red lights decorate store 9
D. Red lights decorate store 8
E. Yellow lights decorate store 2

Suppose that yellow lights decorate exactly two stores on the south side of the street and exactly one store on the north side. If all other conditions remain the same, then which one of the following statements MUST be true?
A. Green lights decorate store 1
B. Red lights decorate store 7
C. Red lights decorate store 10
D. Yellow lights decorate store 2
E. Yellow lights decorate store 8

Which one of the following statements MUST be true?
A. Green lights decorate store 10
B. Red lights decorate store 1
C. Red lights decorate store 8
D. Yellow lights decorate store 8
E. Yellow lights decorate store 10

Six square states having equal area in a country are located in North – South direction in two columns next to each other. States are located in the given order, State 1, State 3, and State 5 are on the western side and State 2, State 4, and State 6 are on the eastern side. Within the six states, there are on the eastern side. Within the six states, there are exactly four medical institutes, two management institutes, and two technical institutes. These eight institutions are located as follows:
No institution is in more than one of the states.
None of the states contains more than one management institute, and none contains more than one technical institute.
None of the states contains both a management institute and a technical institute.
Each management institute is located in a state that contains at least one medical institute.
The technical institutes are located in two states that do not share a common boundary.
State 3 contains a technical institute, and
State 6 contains a management institute.

If one of the states contains exactly two medical institutes and exactly one technical institute, then which combination of three states might contain no medical institute?
A. 1, 3, 5
B. 1, 4, 5
C. 2, 3, 5
D. 2, 4, 6
E. 4, 5, 6

Which one of the following could be true?
A. Sate 1 contains exactly one medical institute
B. Sate 1 contains exactly one technical institute
C. Sate 2 contains exactly one management institute
D. State 5 contains exactly one technical institute
E. State 6 contains exactly one technical institute

A complete and accurate list of the states, any one of which could contain the management institute that is not in State 6, would be _______
A. 1, 4
B. 1, 4, 5
C. 2, 4
D. 1, 2, 4, 5
E. 4, 5

If each of the six states contains at least one of the eight institutions, then which one of the following must be true?
A. There is management institute in state 4
B. There is a management institute in state 1
C. There is a medical institute in state 2
D. There is a medical institute in state 3
E. There is a medical institute is state 4

A, B, C, D, E and F are a group of friends from a club. There are two housewives, one lecturer, one architect, one accountant and one lawyer in the group. There are two married couples in the group. The lawyer is married to D who is a housewife. No lady in the group is either an architect or an accountant. C, the accountant, is married to F who is a lecturer. A is married to D and E is not a housewife.

What is the profession of E?
a. Lawyer
b. Architect
c. Lecturer
d. Accountant

How many members of the group are male?
a. 2
b. 3
c. 4
d. None of these

Each digit, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter A, B, C, D, E, F, G, H and I but not necessarily in this order. Further, each of A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
Determine the value of each of the alphabets such that the above conditions are satisfied.

A survey is conducted in a school among 150 students and it was found that 100 students play cricket, 90 students play football and 80 students play hockey. As physical education was given emphasis in the school, every student had to play at least one sport.
What can be the maximum number of students who play exactly two sports?
What can be the maximum number of students who play exactly one sport?

Prove or Disprove:

There exists some power of 3 which ends in 001.

How many 5 digit numbers are there such that the sum of its digits is odd?
Example: 12345 is a five digit number such that the sum of its digits = 1 + 2 + 3 + 4 + 5 = 15 = an odd number.