How many more digits does the base-3 representation of $987_{10}$ have than the base-8 representation of $987_{10}$?
In order to solve this problem, we must first find the number of digits when $987_{10}$ is converted to each base. Beginning with base-3, we have that $2187>987>729$ or $3^7>987>3^6$. So, we know that the base-3 representation of $987_{10}$ has 7 digits. Similarly with base-8, we have that $4096>987>512$ or $8^4>987>8^3$. So, the base-8 representation of $987_{10}$ has only 4 digits. Therefore, the base-3 equivalent has $7-4=\boxed{3}$ more digits than the base-8 equivalent.
3
Level 3
Number Theory
Problem: How many more digits does the base-3 representation of $987_{10}$ have than the base-8 representation of $987_{10}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many more digits the base-3 representation of 987 has compared to its base-8 representation, we first need to convert the decimal number 987 into both base-3 and base-8.

Starting with base-8, we divide 987 by 8 repeatedly and keep track of the remainders:
1. 987 ÷ 8 = 123 remainder 3
2. 123 ÷ 8 = 15 remainder 3
3. 15 ÷ 8 = 1 remainder 7
4. 1 ÷ 8 = 0 remainder 1

Reading the remainders from bottom to top, the base-8 representation of 987 is 1733. The number of digits in 1733 is 4.

Next, we convert 987 to base-3 using a similar method:
1. 987 ÷ 3 = 329 remainder 0
2. 329 ÷ 3 = 109 remainder 2
3. 109 ÷ 3 = 36 remainder 1
4. 36 ÷ 3 = 12 remainder 0
5. 12 ÷ 3 = 4 remainder 0
6. 4 ÷ 3 = 1 remainder 1
7. 1 ÷ 3 = 0 remainder 1

Reading the remainders from bottom to top, the base-3 representation of 987 is 1102000. The number of digits in 1102000 is 7.

Now, we find the difference in the number of digits between the base-3 and base-8 representations:
Number of digits in base-3: 7
Number of digits in base-8: 4

The difference is calculated as:
7 (base-3 digits) - 4 (base-8 digits) = 3.

Thus, the final answer is 3.<|END_OF_LATENT|> The final answer is $3$.