problem with dumbbells

Question

how do you resolve this problem? I can´t understand.

¿Which one is the less number of dumbbells that that i need to put in the dumbbells plate of a balance to weight 2709kg if you have a dumbbells collection of 1kg , 4kg ,16kg ,64kg,...?

Thanks!

Answer 1

Assuming I haven't misunderstood, all you need to do is write 2709 in base 4. It turns out to be $222111_4$, so you would need 9 such dumbbells.

Answer 2

HINT: Always use the largest dumbbell that doesn’t raise the weight above $2709$ kg. The available dumbbells have weights that are powers of $4$, so the next few are $256,1024$, and $4096$ kg. That last one is too big, so start with a $1024$ kg dumbbell. That leaves $2709-1024=1685$ kg still to be balanced. You can now add another $1024$ kg dumbbell, leaving $1685-1024=661$ kg still to be balanced. Keep going: the biggest dumbbell that will still ‘fit’ is a $256$ kg dumbbell, and when you’ve added it to the pan, only $661-256=405$ kg remains to be balanced. Continue in this fashion, and when you’ve balanced the entire $2709$ kg, see how many dumbbells you’ve used altogether.

Note that the problem is equivalent to writing $2709$ in base four and adding up the digits.