Question

there were between 2200 and 2600 guests invited to a marriage party. the event manager decided that she would arrange the seating so that each table would seat equal no. of guests, and the no. of guests in each table would be an odd number. She worked out that three guests per table
would not work as two guests would be leftover,
neither would five per table, which would leave four guests without a table,
neither would seven per table, which would leave six over, and
nine per table would leave eight left.
However, when she tried seating 11 guests per table it worked exactly, and none were left over. Write down the total number of guests who were invited to the party?​

Answer 1

Given : There were between 2200 and 2600 guests invited to a marriage party. 3 guests per table  would not work as 2 guests would be leftover,

neither would 5 per table, which would leave 4 guests without  a table,  neither would 7 per table, which would leave 6 over, and  9 per table would leave 8 left.  

However, when she tried seating 11 guests per table it worked exactly, and none were left over

To Find : Number of Guests

Solution:

Guest  = G

G = 3A  + 2 = 3A + 3 - 1  =>  3(A + 1)  =  G + 1

G = 5B + 4 = 5B + 5 - 1  =>  5(B + 1)  =  G + 1

G = 7C + 6 = 7C + 7 - 1  =>  7(C + 1)  =   G + 1

G = 9D + 8 = 9D + 9 - 1  =>  9(D + 1)  =  G + 1

G = 11E

G + 1  is LCM or multiple of  3 , 5 , 7 , 9

LCM = 5 *  7  * 9  =  315

G + 1  = 315K

=> G = 315K - 1

G = 11E

=> 11E  = 315K - 1

=> 11E = 11x28K + 7K - 1

=> 11(E - 28K)  = 7K - 1

K = 8

E - 28K = 5

=> E - 28(8) = 5

=> E = 229

G = 11E  = 11 * 229  = 2519

Guest  were between 2200 and 2600

2519 Guest were in Party

other way after  G = 315K - 1

between 2200 and 2600    

2204  = 11 x 200 + 4

2519  =  11 * 229

Hence 2519 Guest

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