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If the units digit of a positive integer n is 2 and there are 4 factor

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Bunuel
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If the units digit of a positive integer n is 2 and there are 4 factor [#permalink] New post  25 Sep 2023, 00:24
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If the units digit of a positive integer n is 2 and there are 4 factors of n that are greater than 1 and less than the square root of n, what is the total number of factors of n?

A. 8
B. 9
C. 10
D. 11
E. Cannot be determined

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chetan2u
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Re: If the units digit of a positive integer n is 2 and there are 4 factor [#permalink] New post  25 Sep 2023, 21:38
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Bunuel wrote:
If the units digit of a positive integer n is 2 and there are 4 factors of n that are greater than 1 and less than the square root of n, what is the total number of factors of n?

A. 8
B. 9
C. 10
D. 11
E. Cannot be determined

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There are FIVE factors including 1 less than \(\sqrt{n}\).
Now, n ends in 2, so it cannot be a perfect square as squares end in 0,1,4,5,6, and 9.

For each factor less than \(\sqrt{n}\), there will be a factor greater than \(\sqrt{n}\)

Thus, 2*5 or 10.
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gmatophobia
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If the units digit of a positive integer n is 2 and there are 4 factor [#permalink] New post  25 Sep 2023, 23:15
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Bunuel wrote:
If the units digit of a positive integer n is 2 and there are 4 factors of n that are greater than 1 and less than the square root of n, what is the total number of factors of n?

A. 8
B. 9
C. 10
D. 11
E. Cannot be determined

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Factors exist in pairs: If \(p\) is a factor less than the \(\sqrt{n}\), then \(\frac{n}{p}\) is a corresponding factor greater or equal to the \(\sqrt{n}\).

\(\frac{n}{p} = p\) ➔ When n is a perfect square, for all other numbers \(\frac{n}{p}\) is greater than \(p\).

Applying this concept to this question. We know that "...there are 4 factors of n that are greater than 1 and less than the square root of n...", therefore the total number of factors can be (5*2) or (5*2)+1

  • If n is a perfect square the total number of factors = 11
  • If n is not a perfect square the total number of factors = 10.

As the number ends in 2, we can infer that the number is not a perfect square. Hence, the number of factors = 10.

Option C
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If the units digit of a positive integer n is 2 and there are 4 factor [#permalink]
25 Sep 2023, 23:15
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