Express 278910 in binary – use the technique of subtracting powers of 2

The decimal number 278910, expressed in binary, is as follows:

               1000100000101111110

Explanation:

Any decimal number can be expressed as a linear combination of powers of 2, as follows:

N = aₙ* 2ⁿ ++ a₀*2⁰, where the coefficients aₓ can be 0 or 1.

This means that any number, can be decomposed in powers of 2, so a useful and at the same time simple way to find the binary equivalent of a decimal number, is simply to substract from the number the maximum power of 2 that gives a positive outcome, and put a "1" in the most left position, filling with zeros to the right till finding the following power of two (obtained repeating the process with the result from the first substraction).

For the first substraction, we try different choices, until we get a positive result substracting 2¹⁸ from 278910, as follows:

278,910-262,144= 16,766.

Intuitively, we know that as being 16 a power of 2, it's possible that a number close to the one we have as a result, be a power of 2 indeed.

Trying with 2¹⁴, we find that we are right, because the result is a small number:  

16,766 - 16,384 = 382

Now. it's very easy, as the greatest power of 2 smaller than 382, is 2⁸=256.  

382-256= 126.

126 can be written as 64+32+16+8+4+2, so all that we need now is, going from left to right, put "1", as the coefficient of the powers of 2 18, 14, 8, 6, 5,4,3,2 and 1, filling with zeros the remaining ones.

The final number can be written as follows:

1000100000101111110

b. xor is the correct answer