An even bit is a binary digit that has a value of 2^2k. An odd bit is a binary digit that is not an even bit. For example, in the binary number 1010101010, the 1's are all odd bits, while the 0's are all even bits.

Prove that if the difference between the number of 1 even bits and the number of 1 odd bits is a multiple of 3, then the binary number itself is a multiple of 3 when converted to decimal.

For example, the binary number 1010100 is a multiple of 3 when converted to decimal since it has three 1 even bits and zero 1 odd bits, whose difference is 3 (which is a multiple of 3).

Question

An even bit is a binary digit that has a value of 2^2k. An odd bit is a binary digit that is not an even bit. For example, in the binary number 1010101010, the 1's are all odd bits, while the 0's are all even bits.

Prove that if the difference between the number of 1 even bits and the number of 1 odd bits is a multiple of 3, then the binary number itself is a multiple of 3 when converted to decimal.

For example, the binary number 1010100 is a multiple of 3 when converted to decimal since it has three 1 even bits and zero 1 odd bits, whose difference is 3 (which is a multiple of 3).

anonymous · Answer

I formulated this question myself; ask if you don't understand anything.