Which one of the following cannot be the square of a natural number?

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30976

75625

28561

143642

answer is D.

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Detailed Solution

Concept- We shall prime factorise each of the provided answers to this question separately. If the prime numbers appear in pairs, it is the square of a natural number, and if the prime numbers do not appear in pairs, it is not the square of natural numbers.
The task is to identify which of the possibilities cannot be the square of a natural integer. This information is provided in the question. We shall prime factorise each of the given numbers separately to determine that. When the prime numbers appear in pairs, we will consider them to be squares of natural numbers; but, if the prime numbers do not appear in pairs, we will not. So let's examine each choice separately.
30976 in Option 1
We'll factor it as follows:
Question Image

So, now that we have all of the prime integers in pairs, we can calculate the square root of 30976 as 176. As a result, choice A is 176 squared, which is a natural number.
2, option 75625.
It can be factored as
Question Image

We can see that all of the prime number pairings are present, and 275 is the square root of 75625. As a result, choice B is 275 squared, a natural number.
28561, Option 3.
Let's factor it as follows:
Question Image

The square root of the number 28561 is 169, therefore we also get the prime numbers in pairs here. Option C is therefore also the square of the natural integer 169.
Choice 4. 143642
We'll factor it as follows:
Question Image

The fact that the elements of the number 143642 are not in pairs is evident from the information presented above. It follows that it cannot be the square of a natural integer.
Hence, the correct answer is option 4.

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