Question

There are a certain number of chairs inside the hall. If the chairs are arranged 5 chairs per row, 7 chairs per row, and 8 chairs per row then 3, 2 and 5 chairs remain respectively. It is known that there are more than 600 chairs and less than 700 chairs inside the hall. Find the number of chairs remaining if the chairs are arranged 13 chairs per row.

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Solution

The correct option is **B**

This question can be solved using Chinese Remainder theorem.

Suppose the no. of chairs is N. It is also known that 600<N<700.

Suppose the quotients when N is divided by 5, 7 and 8 and leaves the remainders 3, 2 and 5 be A, B

and C respectively.

Then, N = 5A+3 = 7B+2 = 8C+5.......................(1)

Solving first for

5A+3=7B+2

We get the first integral solution for (A,B) at (4,3).

The first solution for the above pair of equations = 23.

This increases in an Arithmetic Progression of 35 (LCM of 7 and 5)

Thus the general form of the above equation = 23+35k

Let us equate this to the last part of eqn (1).

8C+5=23+35K

The first integral solution for this occurs at C=11 and K=2

Thus the first number which when divided by 5, 7 and 8; leaves the remainders 3, 2 and 5 is 93.

This number is the first term of an AP with a common difference of 280 (LCM of 7, 5 and 8)

Thus, the number which will lie in the range of 600-700 is 653.

653 when divided by 13 leaves a remainder of 3; which is option (b).

3

This question can be solved using Chinese Remainder theorem.

Suppose the no. of chairs is N. It is also known that 600<N<700.

Suppose the quotients when N is divided by 5, 7 and 8 and leaves the remainders 3, 2 and 5 be A, B

and C respectively.

Then, N = 5A+3 = 7B+2 = 8C+5.......................(1)

Solving first for

5A+3=7B+2

We get the first integral solution for (A,B) at (4,3).

The first solution for the above pair of equations = 23.

This increases in an Arithmetic Progression of 35 (LCM of 7 and 5)

Thus the general form of the above equation = 23+35k

Let us equate this to the last part of eqn (1).

8C+5=23+35K

The first integral solution for this occurs at C=11 and K=2

Thus the first number which when divided by 5, 7 and 8; leaves the remainders 3, 2 and 5 is 93.

This number is the first term of an AP with a common difference of 280 (LCM of 7, 5 and 8)

Thus, the number which will lie in the range of 600-700 is 653.

653 when divided by 13 leaves a remainder of 3; which is option (b).

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