If 45x = 121y, which of the following must be true?

This is a must be true question.

The way the options are presented, we can evaluate I and II first.

Given : 45x = 121y

If both x and y are zero even then the LHS = RHS

Hence Neither I or II can be true.

Let's evaluate III

45x = 121y

Case 1 :

x = 121 & y = 45 ; then yes \(\frac{x}{11}\) is an integer

Case 2 :
However x can well be an fraction and y can adjust accordingly to make the RHS = LHS

Say x = \(\frac{1}{3} \)

45 * x = \(45 * \frac{1}{3}\) = 15

Now y can be \(\frac{15 }{ 121}\) to make LHS = RHS

In this case, \(\frac{x}{11}\) is not an integer.

As it's a must be true question , III can be eliminated.

Option E

Edit : Missed considering zero in the first version. Edited the post to correct the solution.
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If x and y are both zero then 45x = 121y, but (I) x>y and (II) x^2 > y^2 are not true.
IMO Answer is E.

Good catch Aks111! Kudos to you.

P.S - I need to be more careful.
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Official Solution:


If \(45x = 121y\), which of the following must be true?

I. \(x > y\)

II. \(x^2 > y^2\)

III. \(\frac{x}{11}\) is an integer




A. I only
B. II only
C. III only
D. I and III only
E. None of the above


It is important to note that the problem does not specify that \(x\) and \(y\) are necessarily integers, nor does it require \(x\) and \(y\) to be necessarily positive. Therefore, when evaluating each option, it is essential to keep in mind that these variables may take on non-integer or non-positive values!

I. \(x > y\).

This statement is not always true, as \(x\) and \(y\) are not necessarily positive. For instance, consider \(x = -121\) and \(y=-45\), or \(x = 0\) and \(y=0\).

II. \(x^2 > y^2\).

This statement is not always true, as \(x\) and \(y\) are not necessarily integers. For example, consider \(x = \frac{1}{45}\) and \(y=\frac{1}{121}\). Additionally, since \(x\) and \(y\) can both be 0, this statement is also not true in that case.

III. \(\frac{x}{11}\) is an integer.

This statement is also not always true, as \(x\) and \(y\) are not necessarily integers. For instance, consider \(x = \frac{1}{45}\) and \(y=\frac{1}{121}\).


Answer: E
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