Here \(\frac{j^2}{k}\) is odd that is a given so the statement 'E/E can be odd or can be even ' is irrelevant.

\(\frac{j^2}{k}\) is an integer and that too odd. So what are values (even or odd) of k if j is even (given)

You have to go backwards it is already given that the fraction is an odd integer.

Please explain with an example?

Here let us consider the option one at a time:

Option A ::: j and k are both even: It may not be true because if j=1 and k=1 then we can get \(\frac{j^2}{k}= Odd\)

OPtion B::: j = k: it may not be true because if j=5 and k=1; it still satisfy the equation \(\frac{j^2}{k} = Odd\)

Option C ::: If j is even, k is even , this may be true \(\frac{j^2}{k} = odd\) or \(j^2\) = k*Odd, so if j = even then k has to be even to satisfy the equation (Even * Odd = Even)

Option D::: If j is divisible by k this may not be true because if j = 3 , k = 9 , it still satisfy the equation \(\frac{j^2}{k} = Odd\)

Option E ::: \(j^2 > k\) It may not be true, because if j = 1 and k = 1 then \(j^2 = k\) and it still satisfy the equation \(\frac{j^2}{k} = Odd\)

j^2/k is odd implies that both j^2 and k are odd or both are even. So it is like, if j is odd, k is also odd or if k is even then j is even. It cannot be the case that j^2 is odd and k is even or vice versa, because in such a case you will not get an integer.

Now look at the options:
A) j and k are both even - but they can be odd also. so this is not certain and hence out
B) there is no evidence of j and k being equal. so this is not certain and hence out
C) Compare it with option A. Here it is given that j is even and hence k is even, this is exactly what we have concluded above
D) j is divisible by k. Not necessarily. consider j=4 and k=16
E) j^2 > k : this may not always be the case, consider j = 4 and k = 16

So the correct answer is C

Sippose j=4 and k=4 then ?it will be 16/4=4 how come C is the right answer

J^2/K is odd, this has multiple options:
j^2 = k For example (2^2)/4 =1 and 3^3/27=1
J and K are both can be even or odd numbers, as the example above
For this answer b,d,e should be ignored

Answer (A) it is not necessarily true j and k could be odd too.
answer is (C).

Carcass I think the answer should be D because how can j be even for j^2/k to be odd J can never be even, the question asks "Must be true" right, so a statement that's saying "IF" cant be true, isn't this vacuous truth, or am i missing something here.

If j and k are integers and j^2 / k is odd, which of the following must be true ?

Given: \(j\) and \(k\) are integers and \(\frac{j^2}{k}=odd\) --> \(j^2=k*odd\)

(A) j and k are both even: not necessarily true, for example \(j=1\) and \(k=1\);

(B) j = k: not necessarily true, for example \(j=3\) and \(k=1\);

(C) If j is even, k is even: as \(j^2=k*odd\) then in order \(j\) to be even \(k\) must be even too, so this statement must be true;

(D) j is divisible by k: not necessarily true, for example \(j=3\) and \(k=9\);

(E) j^2 > k: not necessarily true, for example \(j=1\) and \(k=1\).

Answer: C.