To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?
I know that the largest power of 2 that is less than or equal to 400 is $2^8 = 256$. If I subtract 256 from 400, I get 144. The largest power of 2 that is less than or equal to 144 is $2^7 = 128$. If I subtract 128 from 144, I get 16. The largest power of 2 that is less than or equal to 16 is $2^4 = 16$. So, I can write $400 = 2^8 + 2^7 + 2^4$. The sum of the exponents of these powers is $8 + 7 + 4 = 19$. This is the least possible sum of the exponents, because if I used smaller powers of 2, I would need more of them to add up to 400, and the sum of the exponents would be larger. For example, if I used $2^6 = 64$ instead of $2^7 = 128$, I would need two more powers of 2 to make up the difference, and the sum of the exponents would be at least $8 + 6 + 6 + 4 = 24$. 