Answer Posted / guest

Prove it using the mathematical induction.

First here are a couple of things to note:

[1] A number whose digits add up to a multiple of three is
divisable by 3.

e.g. 369: 3+6+9=18: 1+8=9 which is a multiple of 3 hence 369
is divisable by 3.

[2] Whenever a number (X) is multiplied with another number
(Y) then the product (X*Y) will have all the factors of X as
well as all the factors of Y in its set of factors.

e.g. if X has factors of (1,P,Q,X) and Y has factors of
(1,Q,R,Y) then X*Y has factors of (1,P,Q,Q,R,X,Y).

Let

N = any series of digits (e.g. N=369)

D = the number of digits in N (e.g. if N=369 then D=3)

P = is a number constructed in the following way : a 1,
followed by (D-1) 0s, followed by another 1, followed by
(D-1) 0s, followed by another 1. (e.g. if N=369 then D=3 and
P would be 1001001) Note that P will always be divisible by 3.

Also, if we multiply N with P we are essentially repeating N
for (D-1) times.

e.g. if N=369 then D=3, P=1001001 and N*P=369369369

Let's start with N=111. It is clear that N is divisible by
3. (From [1])

Also, D=3 and P=1001001

N*P=111111111 (9 times)

The resulting number 111111111 must be divisible by 9 as N
and P both are divisible by 3.

Now, let's start with N=111111111. It is clear that N is
divisible by 9.

Also, D=9 and P=1000000001000000001

N*P=111111111... (27 times)

The resulting number 1111111... (27 times) must be divisible
by 27 as N is divisible by 9 and P is divisible by 3.

Repeat the same procedure for N=1111111... (27 times) The
resulting number 1111111... (81 times) must be divisible by
81 as N is divisible by 27 and P is divisible by 3.

Similarly, for N=1111111... (81 times) The resulting number
1111111... (243 times) must be divisible by 243 as N is
divisible by 81 and P is divisible by 3.

Thus, 1111111... (243 times) is divisible by 243.

In country X, democratic, conservative and justice parties have fought three civil wars in twenty years. TO restore stability an agreement is reached to rotate the top offices President, Prime Minister and Army Chief among the parties so that each party controls one and only one office at all times. The three top office holders must each have two deputies, one from each of the other parties. Each deputy must choose a staff composed of equally members of his or her chiefs party and member of the third party. 1. When Justice party holds one of the top offices, which of the following cannot be true (a) Some of the staff members within that office are justice party members (b) Some of the staff members within that office are democratic party members (c) Two of the deputies within the other offices are justice party members (d) Two of the deputies within the other offices are conservative party members (e) Some of the staff members within the other offices are justice party members. 2. When the democratic party holds presidency, the staff of the prime minister's deputies are composed I. One-fourth of democratic party members II. One-half of justice party members and one-fourth of conservative party members III. One-half of conservative party members and one-fourth of justice party members. (a) I only (b) I and II only (c) II or III but not both (d) I and II or I and III (e) None of these 3. Which of the following is allowable under the rules as stated: (a) More than half of the staff within a given office belonging to a single party (b) Half of the staff within a given office belonging to a single party (c) Any person having a member of the same party as his or her immediate superior (d) Half the total number of staff members in all three offices belonging to a single party (e) Half the staff members in a given office belonging to parties different from the party of the top office holder in that office. 4. The office of the Army Chief passes from Conservative to Justice party. Which of the following must be fired. (a) The democratic deputy and all staff members belonging to Justice party (b) Justice party deputy and all his or hers staff members (c) Justice party deputy and half of his Conservative staff members in the chief of staff office (d) The Conservative deputy and all of his or her staff members belonging to Conservative party (e) No deputies and all staff members belonging to conservative parties.

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