If (P)g represents the number P in base — 8, then (144) g+ (175)g=

The correct answer is (341)₈.

We need to add the two octal (base-8) numbers (144)₈ and (175)₈. Octal addition works just like decimal addition, except that we carry over when a column sum reaches 8 (instead of 10).

Let's set up the addition column by column, starting from the rightmost digit (ones place) and moving left.

Step 1: Add the Ones (Units) column
Digits: 4 (from 144) + 5 (from 175) = 9 in decimal.
Since 9 ≥ 8, we divide: 9 = 1 × 8 + 1
So we write down 1 and carry 1 to the next column.

Step 2: Add the Eights (Middle) column
Digits: 4 (from 144) + 7 (from 175) + 1 (carry) = 12 in decimal.
Since 12 ≥ 8, we divide: 12 = 1 × 8 + 4
So we write down 4 and carry 1 to the next column.

Step 3: Add the Sixty-fours (Leftmost) column
Digits: 1 (from 144) + 1 (from 175) + 1 (carry) = 3 in decimal.
Since 3 < 8, no carry needed.
So we write down 3.

Summary of the column-by-column addition:

$$\begin{array}{rrrr}\text{Carry:}& 1& 1& \\ & 1& 4& 4\\ +& 1& 7& 5\\ =& 3& 4& 1\end{array}$$

Verification using Decimal Conversion:
We can double-check by converting both octal numbers to decimal, adding, then converting back.

$$(144){}_8=1\times 8^2+4\times 8^1+4\times 8^0=64+32+4=100$$

$$(175){}_8=1\times 8^2+7\times 8^1+5\times 8^0=64+56+5=125$$

Sum in decimal: 100 + 125 = 225

Now convert 225 back to octal:
225 ÷ 8 = 28 remainder 1
28 ÷ 8 = 3 remainder 4
3 ÷ 8 = 0 remainder 3

Reading the remainders from bottom to top gives: 341

Therefore:

$$(144){}_8+(175){}_8=(341){}_8$$