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Is y an integer? (1) y^3 is an integer (2) 3y is an integer [#permalink]

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10 Feb 2012, 07:07

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Is y an integer?

(1) y^3 is an integer --> y is either an integer (..., -1, 0, 1, 2, ...) or \(\sqrt[3]{integer}\), for example \(\sqrt[3]{2}\). Not sufficient. Notice here that y cannot be some reduced fraction like 1/3 or 13/5, because in this case y^3 won't be an integer.

(2) 3y is an integer --> y is either an integer or integers/3, for example: 1/3, 2/3, ... Not sufficient. Notice here that y cannot be some irrational number like \(\sqrt{2}\) or \(\sqrt[3]{integer}\), because in this case 3y won't be an integer.

(1)+(2) Since both y^3 and 3y are integers then, as discussed above, y must be an integer. Sufficient.

(1) y^3 is an integer --> y is either an integer (..., -1, 0, 1, 2, ...) or \(\sqrt[3]{integer}\), for example \(\sqrt[3]{2}\). Not sufficient. Notice here that y can not be some reduced fraction like 1/3 or 13/5, because in this case y^3 won't be an integer.

B) 3y is an integer --> y is either an integer or integers/3, for example: 1/3, 2/3, ... Not sufficient. Notice here that y can not be some irrational number like \(\sqrt{2}\) or \(\sqrt[3]{integer}\), because in this case 3y won't be an integer.

(1)+(2) Since both y^3 and 3y are integers then, as discussed above, y must be an integer. Sufficient.

Answer: C.

Bunnel can you explain here by putting in some values for y. I am still unable to trace how by using both I can find that y is an integer.

(1) y^3 is an integer --> y is either an integer (..., -1, 0, 1, 2, ...) or \(\sqrt[3]{integer}\), for example \(\sqrt[3]{2}\). Not sufficient. Notice here that y can not be some reduced fraction like 1/3 or 13/5, because in this case y^3 won't be an integer.

B) 3y is an integer --> y is either an integer or integers/3, for example: 1/3, 2/3, ... Not sufficient. Notice here that y can not be some irrational number like \(\sqrt{2}\) or \(\sqrt[3]{integer}\), because in this case 3y won't be an integer.

(1)+(2) Since both y^3 and 3y are integers then, as discussed above, y must be an integer. Sufficient.

Answer: C.

Bunnel can you explain here by putting in some values for y. I am still unable to trace how by using both I can find that y is an integer.

Sure. Generally \(\sqrt[3]{integer}\) is either an integer itself or an irrational number, it can not be some reduced fraction like 1/2 or 2/3. (The same way as \(\sqrt{integer}\) is either an integer itself or an irrational number).

From (1) \(y=integer\) or \(y=\sqrt[3]{integer}\);

From (2) \(y=integer\) or \(y=\frac{integer}{3}\);

So, from (1)+(2) \(y=integer\).

Because if from (1) \(y=\sqrt[3]{integer}\), for example if \(y=\sqrt[3]{2}\), then \(3y=integer\) won't hold true for (2): \(3y={3*\sqrt[3]{2}}\neq{integer}\). The same way: if from (2) \(y=\frac{integer}{3}\), for example if \(y=\frac{1}{3}\), then \(y^3=integer\) won't hold true for (1): \(y^3=(\frac{1}{3})^3\neq{integer}\).

(1) y^3 is an integer --> y is either an integer (..., -1, 0, 1, 2, ...) or \(\sqrt[3]{integer}\), for example \(\sqrt[3]{2}\). Not sufficient. Notice here that y can not be some reduced fraction like 1/3 or 13/5, because in this case y^3 won't be an integer.

B) 3y is an integer --> y is either an integer or integers/3, for example: 1/3, 2/3, ... Not sufficient. Notice here that y can not be some irrational number like \(\sqrt{2}\) or \(\sqrt[3]{integer}\), because in this case 3y won't be an integer.

(1)+(2) Since both y^3 and 3y are integers then, as discussed above, y must be an integer. Sufficient.

Answer: C.

Bunnel can you explain here by putting in some values for y. I am still unable to trace how by using both I can find that y is an integer.

Sure. Generally \(\sqrt[3]{integer}\) is either an integer itself or an irrational number, it can not be some reduced fraction like 1/2 or 2/3. (The same way as \(\sqrt{integer}\) is either an integer itself or an irrational number).

From (1) \(y=integer\) or \(y=\sqrt[3]{integer}\);

From (2) \(y=integer\) or \(y=\frac{integer}{3}\);

So, from (1)+(2) \(y=integer\).

Because if from (1) \(y=\sqrt[3]{integer}\), for example if \(y=\sqrt[3]{2}\), then \(3y=integer\) won't hold true for (2): \(3y={3*\sqrt[3]{2}}\neq{integer}\). The same way: if from (2) \(y=\frac{integer}{3}\), for example if \(y=\frac{1}{3}\), then \(y^3=integer\) won't hold true for (1): \(y^3=(\frac{1}{3})^3\neq{integer}\).

Hope it's clear.

Unable to understand from the explanation provided...... how from (1) + (2) --> y=Integer ??? Can you pls provide some alternate solution/explanation.

Re: Is y an integer? (1) y^3 is an integer (2) 3y is an integer [#permalink]

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02 Jan 2014, 11:23

subhajeet wrote:

Is y an integer?

(1) y^3 is an integer (2) 3y is an integer

Statement I is insufficient

y ^ 3 = 1 y = 1 (YES y is an integer) y ^ 3 = 2 y = 2^1/3 (y is not an integer)

Statement II is insufficient 3y = 3 y = 1 (YES y is an integer) 3y = 1 y = 1/3 (y is not an integer)

Combining is sufficient (Usually your approach should be algebraic here) y^3 = p 3y = q

If 3y = q then the only problem which makes y not an integer is that y is a fraction which is ruled out by the first statement as Fraction ^ 3 can never be an integer. Similarly the number being a cube root (problem in the first statement) is ruled out by the second statement as 3(Surd) cannot be an integer.

Re: Is y an integer? (1) y^3 is an integer (2) 3y is an integer [#permalink]

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05 Jan 2016, 15:05

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