Math / Numbers & Operations

QuestionWrite 313 in the base-six system. $313=$ $\square$

Studdy Solution

STEP 1

What is this asking? Convert the number 313 from base ten (our usual number system) to base six. Watch out! Don't forget that base six only uses digits from 0 to 5.

STEP 2

1. Find the largest power of six
2. Divide and conquer
3. Write the base-six number

STEP 3

First, we need to find the largest power of six that fits into 313.
Let's calculate the powers of six:
$6^0 = 1, \quad 6^1 = 6, \quad 6^2 = 36, \quad 6^3 = 216, \quad 6^4 = 1296$ Since $1296$ is too large, the largest power of six that fits into 313 is $6^3 = 216$ .

STEP 4

Now, let's see how many times $6^3 = 216$ fits into 313.
We do this by dividing:
$313 \div 216 \approx 1.45$ Since we can only use whole numbers in our base-six digits, we take the whole number part, which is 1.
This means $6^3$ fits 1 time.

STEP 5

Subtract $1 \cdot 216$ from 313 to see what's left:
$313 - 216 = 97$

STEP 6

Next, let's see how many times $6^2 = 36$ fits into 97:
$97 \div 36 \approx 2.69$ So, $6^2$ fits 2 times.
Subtract $2 \cdot 36$ from 97:
$97 - 72 = 25$

STEP 7

Now, check how many times $6^1 = 6$ fits into 25:
$25 \div 6 \approx 4.16$ So, $6^1$ fits 4 times.
Subtract $4 \cdot 6$ from 25:
$25 - 24 = 1$

STEP 8

Finally, $6^0 = 1$ fits into 1 exactly 1 time.
Subtract $1 \cdot 1$ from 1:
$1 - 1 = 0$

STEP 9

Now, let's put it all together!
We found:
- $6^3$ fits 1 time - $6^2$ fits 2 times - $6^1$ fits 4 times - $6^0$ fits 1 time
So, the base-six representation of 313 is 1241.

STEP 10

The number 313 in base ten is written as 1241 in base six.

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