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Convert 21 from decimal to octal

(base 8) notation:

Power Test

Raise our base of 8 to a power

Start at 0 and increasing by 1 until it is >= 21

8⁰ = 1

8¹ = 8

8² = 64 <--- Stop: This is greater than 21

Since 64 is greater than 21, we use 1 power less as our starting point which equals 1

Build octal notation

Work backwards from a power of 1

We start with a total sum of 0:

8¹ = 8

The highest coefficient less than 7 we can multiply this by to stay under 21 is 2

Multiplying this coefficient by our original value, we get: 2 * 8 = 16

Add our new value to our running total, we get:
0 + 16 = 16

This is <= 21, so we assign our outside coefficient of 2 for this digit.

Our new sum becomes 16

Our octal notation is now equal to 2

8⁰ = 1

The highest coefficient less than 7 we can multiply this by to stay under 21 is 5

Multiplying this coefficient by our original value, we get: 5 * 1 = 5

Add our new value to our running total, we get:
16 + 5 = 21

This = 21, so we assign our outside coefficient of 5 for this digit.

Our new sum becomes 21

Our octal notation is now equal to 25

Final Answer