If n and y are positive integers and 450 y = nandsup3;, which of the following must be an integer?a)b)c)d)None of theseCorrect answer is option 'A'. Can you explain this answer?

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If n and y are positive integers and 450 y = n³, which of the following must be an integer?

a)
b)
c)
d)
None of these

Correct answer is option 'A'. Can you explain this answer?

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If n and y are positive integers and 450 y = n³, which of the fol...

450 y = n³ implies that 450 y is the cube of an integer.

When we prime-factorize the cube of an integer, we get 3 (or a multiple of 3) of every prime factor.

8 is the cube of an integer because 8 = 2³ = 2*2*2.

Thus, when we prime-factorize 450 y, we need to get at least 3 of every prime factor.

Here's the prime factorization of 450 y:

450 y = 2 * 3² * 5² * y

Since 450 provides only one 2, two 3's, and two 5's, and we need at least 3 of every prime factor, the missing prime factors must be provided by y. Thus, y must provide at at least two more 2's, one more 3, and one more 5.

Smallest possible case:

y = 2² * 3 * 5

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