The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repetition of digits allowed) is equal to:

a1a2a3
Number of numbers 5 × 3 × 5 = 75
a4
a1a2a3
2 ways for a4
Number of number = 2 × 5 × 3 × 5 = 150
Required number = 75 + 2 × 75 = 225
Let's consider the number of 4 digit numbers less than 7000.
The first digit can be 1, 3, 7 (3 choices)
The second, third and fourth digit can be any of 0, 1, 3, 7, 9 (5 choices each)
Number of such numbers = 3 × 5 × 5 × 5 = 375
Now consider 3 digit numbers:
First digit can be 1, 3, 7, 9 (4 choices)
Second and third digits can be any of 0, 1, 3, 7, 9 (5 choices each)
Number of such numbers = 4 × 5 × 5 = 100
Now consider 2 digit numbers:
First digit can be 1, 3, 7, 9 (4 choices)
Second digit can be any of 0, 1, 3, 7, 9 (5 choices)
Number of such numbers = 4 × 5 = 20
Now consider 1 digit numbers:
Number of such numbers = 4 (1,3,7,9)
Total number of natural numbers = 375 + 100 + 20 + 4 = 499. There is a mistake in the above approach.
Let's consider the number of 4 digit numbers less than 7000. The first digit can be 1, 3, or 7 (3 choices). The remaining digits can be any of 0, 1, 3, 7, 9 (5 choices each). So there are 3 × 5 × 5 × 5 = 375 such numbers. For 3-digit numbers, there are 5 choices for each digit, resulting in 5 × 5 × 5 = 125 numbers. For 2-digit numbers, there are 5 × 5 = 25 numbers. For 1-digit numbers, there are 4 (1, 3, 7, 9). Thus, the total is 375 + 125 + 25 + 4 = 529. There is still something wrong here. Let's try another approach.
For 1-digit numbers: 4
For 2-digit numbers: 55 = 25
For 3-digit numbers: 555 = 125
For 4-digit numbers less than 7000: 3555 = 375
Total: 4 + 25 + 125 + 375 = 529. There must be a mistake in the question or the provided solution.