Question

Mahesh works as a manager in a hotel. He has to arrange seats in the hall for a function. A hall
has a certain number of chairs. Guests want to sit in different groups like in pairs, triplets,
quadruplets, fives, and sixes, etc. When Mahesh arranges chairs in such a pattern as in 2’s,
3’s, 4’s 5’s, and 6’s then 1, 2, 3, 4, and 5 chairs are left respectively. But when he arranges in
11’s, no chair will be left.


(i) In the hall, how many chairs are available?
(a) 407 (b) 143
(c) 539 (d) 209
(ii) If one chair is removed, which arrangements are possible now?
(a) 2 (b) 3
(c) 4 (d) 5
(iii)If one chair is added to the total number of chairs, how many chairs will be left when
arranged in 11’s.
(a) 1 (b) 2
(c) 3 (d) 4
(iv) How many chairs will be left in the original arrangement if the same number of chairs will be
arranged in 7’s?
(a) 0 (b) 1
(c) 2 (d) 3
(v) How many chairs will be left in the original arrangement if the same number of chairs will be
arranged in 9’s?
(a) 8 (b) 1
(c) 6 (d) 3

Answer 1

Step-by-step explanation:

(i)

Let us solve the problem by checking the given options.

Option (a) 407

• When divided by 2, leaves remainder 1
• When divided by 3, leaves remainder 2
• When divided by 4, leaves remainder 3
• When divided by 5, leaves remainder 2, and this does not match our condition

Option (b) 143

• When divided by 2, leaves remainder 1
• When divided by 3, leaves remainder 2
• When divided by 4, leaves remainder 3
• When divided by 5, leaves remainder 3, and this does not match our condition

Option (c) 539

• When divided by 2, leaves remainder 1
• When divided by 3, leaves remainder 2
• When divided by 4, leaves remainder 3
• When divided by 5, leaves remainder 4
• When divided by 6, leaves remainder 5
• When divided by 11, leaves remainder 0
• Thus 539 matches all the given conditions.

Option (d) 209

• When divided by 2, leaves remainder 1
• When divided by 3, leaves remainder 2
• When divided by 4, leaves remainder 1, and this does not match our condition

Option (c) 539 is correct.

(ii)

From 539 chairs, if one chair is removed, (539 - 1) = 538 chairs will be left.

• 538 leaves remainder 0 when divided by 2
• 538 leaves remainder 1 when divided by 3
• 538 leaves remainder 2 when divided by 4
• 538 leaves remainder 3 when divided by 5

Option (a) 2 is correct.

(iii)

If one chair is added with 539 chairs, we will have (539 + 1) = 540 chairs.

540, when divided by 11, leaves remainder 1

Option (a) 1 is correct.

(iv)

539, when divided by 7, leaves remainder 0

Option (a) 0 is correct.

(v)

539, when divided by 9, leaves remainder 8

Option (a) 8 is correct.