Calculation?

Question

Professor Snape owns a balance scale that he uses to weight the potion samples. He always places the samples on the left pan of the balance and places weights (can be any value) on the right pan until a balance is achieved. What is the smallest number of weight he will need if he wishes to be able to balance all possible samples weighing an integral number of grams between
(a)1 and 1000?

(b)1 and n?

Answer 1

a) Usually is used ('easy' numbers)
1 2 2 5
10 20 20 50
100 200 200 500 grams
This is 3 x 4 = 12 weights.

However it can with less weights using
1 2 4 8 16 32 64 128 256 and 512.
This is only 9 weights. This is the minimum.

b) In general the minimal numbers of weights is the integer of
log(1000) / log(2) (any base)

Th

Answer 2

Prime factors.