Base prime representation of a natural number is defined using the exponents of its prime factorization as follows. Each place in a base prime represents a prime number, and it is occupied by the corresponding exponent of that prime, starting on the right side with the smallest prime number and proceeding to the left with the next largest prime number. For instance, since $84 = 7^1 \times 5^0 \times 3^1 \times 2^2$, then $84$ would be written as $1012$ in base prime. What is $225$ written in base prime?
The prime factorization of $225$ is $225 = 15^2 = 3^2 \times 5^2$. Since $2$ does not divide into $225$, we treat $2$ as having a $0$ exponent; the next two primes are $3$ and $5$. Thus, the answer is $\boxed{220}.$
220
Level 4
Number Theory
Problem: Base prime representation of a natural number is defined using the exponents of its prime factorization as follows. Each place in a base prime represents a prime number, and it is occupied by the corresponding exponent of that prime, starting on the right side with the smallest prime number and proceeding to the left with the next largest prime number. For instance, since $84 = 7^1 \times 5^0 \times 3^1 \times 2^2$, then $84$ would be written as $1012$ in base prime. What is $225$ written in base prime?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the base prime representation of 225, first determine its prime factorization. The prime factorization of 225 is calculated as follows:

1. Start by dividing 225 by the smallest prime number, which is 2. Since 225 is odd, it is not divisible by 2.
2. Next, try dividing by 3. Since 225 divided by 3 equals 75, we have one factor: 3.
3. Continue factoring 75. Dividing 75 by 3 again gives 25, so we have another factor of 3.
4. Now factor 25. The smallest prime that divides 25 is 5. Dividing 25 by 5 gives 5, and dividing 5 by 5 gives 1, resulting in another factor of 5.

Thus, the complete prime factorization of 225 is:
225 = 3^2 × 5^2.

Next, identify the primes involved in the factorization. The primes are 2, 3, and 5. Since 2 is not a factor of 225, its exponent will be 0. The exponents of the primes in the factorization are:
- For 2: 0
- For 3: 2
- For 5: 2

Now, arrange these exponents according to the order of the primes, starting from the smallest prime (2) to the largest prime (5). The base prime representation is constructed as follows:
- The exponent for 2 is 0, so the first digit is 0.
- The exponent for 3 is 2, so the second digit is 2.
- The exponent for 5 is 2, so the third digit is 2.

Combining these digits gives the base prime representation of 225 as 220.<|END_OF_LATENT|> The final answer is $220$.