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13 Jun 2025, 06:53
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E
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If the product of √x and √y is an integer and the total number of factors of y/3 is odd, which of the following must be true?
I. The total number of factors of x is odd.
II. The total number of factors of x is even.
III. The total number of factors of 3x is odd.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
I. The total number of factors of x is odd.
II. The total number of factors of x is even.
III. The total number of factors of 3x is odd.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
13 Jun 2025, 13:43
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siddhantvarma
Since, number of factors of \(\frac{y}{3}\) are odd => y = (perfect square)*3
Therefore, \(\sqrt{y}\) = \(\sqrt{(perfect square)*3}\) = \(integer * \sqrt{3}\)
Now, for \(\sqrt{x} * \sqrt{y}\) to be an integer, x needs to have one 3 with any perfect square so that when \(\sqrt{x}\) and \(\sqrt{y}\) are multiplied the \(\sqrt{3}\) of \(\sqrt{y}\) gets multiplied with \(\sqrt{3}\) of \(\sqrt{x}\) and the square root goes away giving us an integer.
So, x = 3 * (any perfect square).
Now, we look at the statements,
I. Since x = 3 * (any perfect square), the number of factors of x will be a multiple of 2 (\(3^1\) => 1+1 = 2); it cannot be odd.
II. As we proved above, total number of factors of x has to be even.
III. 3x = \(3^2\) * (any perfect square), the number of factors will be 3 * any odd number = odd number. So this also has to be true.
Answer E.
13 Jun 2025, 20:25
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siddhantvarma
Taking an example will further clarify things.
Only perfect squares have ODD total # of factors .
\(\frac{y}{3}\)= perfect square.
Thus let \(y\) = \(4*3 \)
\( \sqrt{y}= \sqrt{4*3} = 2\sqrt 3\)
Let \(x= 27\) thus \( \sqrt{x}= \sqrt{9*3} = 3\sqrt{3} \). Because \( \sqrt{x} \) must also be of the form \(p\sqrt{3}\) (\(\)p is also an integer) to get \(\sqrt{x}*\sqrt{y} = \) integer.
Now checking for integer = \( \sqrt{x}* \sqrt{y}= 3\sqrt 3* 2\sqrt{3} = 6*3 = 18\) = integer
Total factors of \(x\) is \(4.\) (\( x= 3^3 \))
1.The total number of factors of \(x\) is odd = False
2.The total number of factors of \(x\) is even= True
3.The total number of factors of \(3x\) is odd.= \(3*3^3 =3^4\) Thus total factors are \(5\) = True
II and III must be true.
Ans E
Hope it helped.
