To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?
To express 1562 as a sum of distinct powers of 2, I need to find the largest power of 2 that is less than or equal to 1562, and then subtract it from 1562 and repeat the process until I get 0. The largest power of 2 that is less than or equal to 1562 is $2^{10} = 1024$. So 1562 can be written as $2^{10} + (1562 - 1024) = 2^{10} + 538$. Now I need to express 538 as a sum of distinct powers of 2. The largest power of 2 that is less than or equal to 538 is $2^9 = 512$. So 538 can be written as $2^9 + (538 - 512) = 2^9 + 26$. Now I need to express 26 as a sum of distinct powers of 2. The largest power of 2 that is less than or equal to 26 is $2^4 = 16$. So 26 can be written as $2^4 + (26 - 16) = 2^4 + 10$. Now I need to express 10 as a sum of distinct powers of 2. The largest power of 2 that is less than or equal to 10 is $2^3 = 8$. So 10 can be written as $2^3 + (10 - 8) = 2^3 + 2$. Now I need to express 2 as a sum of distinct powers of 2. The only power of 2 that is equal to 2 is $2^1$. So 2 can be written as $2^1$. Therefore, 1562 can be written as a sum of distinct powers of 2 as $2^{10} + 2^9 + 2^4 + 2^3 + 2^1$. The sum of the exponents of these powers is $10 + 9 + 4 + 3 + 1 = 27$. 