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# base number

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A certain integer has 4 digits when written in base 8. The same integer has d digits when written in base 3. What is the sum of all possible values of d?

Dec 16, 2020

### Best Answer

#1
+1

We know that the integer has four digits when written in base 8. This means that the smallest number it could possibly be is 1000 base 8 and the largest number it could be is 7777 base 8 (base 8 only has digits 0-7).

1000 base 8 is 512, and 7777 base 8 is (10000 base 8) - 1 = 4095. Therefore there are 4095 - 512 + 1 = 3584 possible integers that could be used for the problem. However, this is not what the problem is looking for.

We are looking for the number of digits any of the 3584 possible numbers has in base 3. The smallest integer, 512, is 200222 base 3. This has 6 digits, so in this case d is 6. I will not bother to convert 4095 base 10 to base 3 because I know that the highest power of 3 that goes into it is 3^7. This means the number will have 8 digits, and d is 8.

Since the highest value of d is 8 and the lowest value of d is 6, the sum of the possible values of d is:

6+7+8=21.

Dec 16, 2020

### 1+0 Answers

#1
+1
Best Answer

We know that the integer has four digits when written in base 8. This means that the smallest number it could possibly be is 1000 base 8 and the largest number it could be is 7777 base 8 (base 8 only has digits 0-7).

1000 base 8 is 512, and 7777 base 8 is (10000 base 8) - 1 = 4095. Therefore there are 4095 - 512 + 1 = 3584 possible integers that could be used for the problem. However, this is not what the problem is looking for.

We are looking for the number of digits any of the 3584 possible numbers has in base 3. The smallest integer, 512, is 200222 base 3. This has 6 digits, so in this case d is 6. I will not bother to convert 4095 base 10 to base 3 because I know that the highest power of 3 that goes into it is 3^7. This means the number will have 8 digits, and d is 8.

Since the highest value of d is 8 and the lowest value of d is 6, the sum of the possible values of d is:

6+7+8=21.

Guest Dec 16, 2020