1. We need to count all natural numbers less than $7000$ that can be formed using the digits $\{0,1,3,7,9\}$ with repetition allowed.

2. Since the number must be less than $7000$, possible lengths are:

• $1$-digit numbers
• $2$-digit numbers
• $3$-digit numbers
• $4$-digit numbers less than $7000$

Also, the first digit cannot be $0$.

1. One-digit numbers

Possible digits are $1,3,7,9$.

So, number of $1$-digit numbers: $4$

2. Two-digit numbers

• First digit: cannot be $0$, so choices are $1,3,7,9$ rightarrow $4$ choices
• Second digit: any of $0,1,3,7,9$ rightarrow $5$ choices

Hence, $4 \times 5 = 20$

3. Three-digit numbers

• First digit: $4$ choices ($1,3,7,9$)
• Second digit: $5$ choices
• Third digit: $5$ choices

Hence, $4 \times 5 \times 5 = 100$

4. Four-digit numbers less than $7000$

The thousand's digit must be less than $7$, and also cannot be $0$. From the set $\{0,1,3,7,9\}$, valid first digits are only: $1,3$ So, first digit has $2$ choices.

Each of the remaining three digits can be any of the $5$ given digits.

Hence, $2 \times 5 \times 5 \times 5 = 250$

5. Total count

Adding all cases: $4 + 20 + 100 + 250 = 374$

So, the required number of natural numbers is $\boxed{374}$

6. Option check

• A: $374$  Correct
• B: $372$  Incorrect
• C: $375$  Incorrect
• D: $250$  Incorrect

Therefore, the correct option is A.