A group of 17 cat herders purchase some number of cats and t

A group of 17 cat herders purchase some number of cats, and they arrive as 11 equally sized herds and one extra herd of 6 cats. When they divide the cats up between themselves into 17 equal groups, 3 are left over and must be released to roam the land. What is the smallest number of cats they can have?

Solution

Let the number of cats in each group of 11 \"equally sized herds\" be \'x\' cats.

Let the number of cats finally with each group be \'y\' cats.

Now,

Number of cats they got to purchase are : 11x+6................(1)

These cats are divided into 17 groups and 3 are left over. This implies number of cats are: 17y+3 (.............(2)

But, (1)=(2)... as number of cats is the same.

11x+6 = 17y+3

--> 11x+3=17y

==> y= (11x+3)/17

==> y is the no. of cats each group has finally which is a whole number.

This implies even (11x+3)/17 should be a whole number.

For it to be a whole number (11x+3) should be exactly divisible by 17.

Now put values into x

if x=1, 11x+3= 14.............. Not divisible by 17

if x=2, 11x+3= 25.............. Not divisible by 17.... and so on

if x=9, 11x+3=102............. divisible by 17

Therfore total no. of cats is 102+3( they left for free)

==> total cats=105