Rewrite the given proof to demonstrate how and why block con

Rewrite the given proof to demonstrate how and why block conversions from octal to binary works (this requires a rigorous proof).

We have learned that we can block-convert a binary number to octal by grouping the binary number into blocks of 3 digits (going from right to left, perhaps padding the binary number with one or two leading zeros to complete the leftmost block of three) and converting each 3 digit binary number into one octal digit. The following is a proof that this procedure works as advertised Suppose n is a nonnegative integer and its binary expansion is given by k=0 where each dk E 10,1) and m is a nonnegative integer. We can assume without loss of generality that the number of terms in this sum is a multiple of 3,i.6e m + 1 = 3q for some natural number q We now group the sum into blocks of 3 terms each, as follows 0-1 2 3i 3i+j 3i+ j 1:0 j=0 We now set o; -d3i + 2d3i+1 + 4d3i+2 for all i and get Since each dk is 0 or 1, the o; satisfy 0 S oi S 7, i.e., there are octal digits. We have found the octal expansion of n, and it is obtained by block-converting three binary digits at a time to octal, from right to left.

Solution

here we have to conver 3 digit binary number into one octal digit

and here m is number of digits which is in \"n\".

the number of terms in this sum is a multiple of 3 i.e m + 1 = q for some natural number q.

and n is divided into three blocks then we find sum of every block and add the blocks.

then the outer sum = sum of all blocks

and the inner sum = sum of individual block