Let

x=no. Of shuttlecocks

y= no. of cricket balls

Then eqn 8x+15y=769

Dividing eqn by 8 we get

Rem(8x+15y/8)=rem(769/8)

=>Rem(7y/8)=1

By trial ,y=7 satisfies the above condition and the other values of y can be obtained by adding 8 successively

y=7,15,23,31,39,47,55

But for y>=55,15y>769.

So y <55

The corresponding values of x are 83 ,68,53,38,23,8.

Hence the irems can be purchased in 6.