If 321 base 4 is divided by 23 base 4 and leaves a remainder 'R' what is the value of 'R' ?

To find the value of the remainder when dividing 321 base 4 by 23 base 4, follow these steps:

Step 1: Convert the numbers to decimal form.
To convert a number from base 4 to base 10, multiply each digit of the number by the appropriate power of 4 and sum them up.

321 base 4 = (3 × 4^2) + (2 × 4^1) + (1 × 4^0)
= (3 × 16) + (2 × 4) + (1 × 1)
= 48 + 8 + 1
= 57

23 base 4 = (2 × 4^1) + (3 × 4^0)
= (2 × 4) + (3 × 1)
= 8 + 3
= 11

Now we can rewrite the problem as: 57 ÷ 11 with a remainder 'R'.

Step 2: Perform the division.
57 ÷ 11 = 5 remainder 2

Step 3: Convert the remainder to base 4.
To convert the remainder (2 in decimal) to base 4, divide it by 4 repeatedly and collect the remainders in reverse order.

2 ÷ 4 = 0 remainder 2

The remainder 'R' when dividing 321 base 4 by 23 base 4 is 2.

or:

321base 4 = 3(4^2) + 2(4) + 1
= 48 + 8 + 1 = 57 in base 10

23base 4 = 2(4) + 3 = 11 in base 10

57/11 = 5 2/11
so the remainder in base 10 is 2
and since 2base 4 = 2base 10

R = 2

Steve's method is more direct, however you have to know how to do arithmetic in base 4.

or:

321base 4 = 3(4^2) + 2(4) + 1
= 48 + 8 + 1 = 57 in base 10

23base 4 = 2(4) + 3 = 11 in base 10

57/11 = 5 2/11
so the remainder in base 10 is 2
and since 2base 4 = 2base 4

R = 2

Steve's method is more direct, however you have to know how to do arithmetic in base 4.

R=2

23*11 base 4 is

023
230
-----
313
+ 2
------
321