Final answer:
The smallest positive perfect square that is divisible by the four smallest primes (2, 3, 5, 7) is 44100, because it is the square of the product of these primes. None of the options provided (4, 9, 16, 25) meet this condition, indicating a potential error in the question or options.
Step-by-step explanation:
The student asks: What's the smallest positive perfect square that's divisible by the four smallest primes?
To find the smallest positive perfect square divisible by the four smallest primes, we need to consider the four smallest prime numbers first, which are 2, 3, 5, and 7. In order for a number to be divisible by all of these primes, it must be a multiple of their product. The product of these primes is 2 x 3 x 5 x 7 = 210. However, since we are looking for a perfect square, we need to ensure that each prime factor is included an even number of times in the prime factorization of the number in question.
Therefore, the smallest positive perfect square divisible by all four smallest primes is 2^2 x 3^2 x 5^2 x 7^2 = 44100. When we take the square root, √44100 = 210, which confirms that 44100 is indeed a perfect square.
Comparing options a through d given in the question:
- (a) 4 = 2^2
- (b) 9 = 3^2
- (c) 16 = 2^4
- (d) 25 = 5^2
None of these options represent the correct answer since 44100 is not listed. The correct answer would be one that includes the prime factors of 210 squared to satisfy the condition, which isn't available in the given options, implying that there might be an error in the question or options provided.