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

If j and k are integers and \(\frac{{j^2}}{k}\) is odd, which of the following must be true?

(A) j and k are both even
(B) j = k
(C) If j is even, k is even
(D) j is divisible by k
(E) \(j^2\) > k

What's the best way to approach these questions? Picking numbers or anything else?

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.
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Think that if \(j^2/k =\) Odd Integer then j^2 is divisible by k. Also either both of them are odd or both j and k are even.
It is not possible that one of them is odd and one of them is even. Why?
If j is even and k is odd, j^2/k is even since j^2 will retain its 2s.
If k is even and j is odd, j^2/k is not an integer because an odd number is never divisible by an even number.
So if j is even, k has to be even.

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

Actually you can use mixed approach.

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.
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Hi Bunuel, please change format of the question, when I saw the question, I thought J to the power of 2/K and started to solve.

When I saw your solution, then I realized that I read the question in different way.Such typing mistakes must be avoided.

j^2/k mathematically can only mean j^2 divided by k. If it were j to the power of k/2 it would be written as j^(2/k).
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Let’s check each answer choice using numbers.

(A) j and k are both even

Let’s say j = 3 and k = 1: we have 3^2/1 = 9, but neither j nor k is even. So, A is not the answer.

(B) j = k

Let’s say j = 3 and k = 1: we have 3^2/1 = 9, but j is not equal to k. So, B is not the answer.

(C) If j is even, k is even

This is an “if-then” statement. The only way this is false is if we can find an even number j and an odd number k and still have j^2/k = odd. However, if j is even, then j^2 is even. If j^2 is even and k is odd, and if j^2 is divisible by k, then j^2/k must be even since even/odd can’t ever be odd. So, we can’t find any even number j and odd number k such that j^2/k = odd, and thus C must be the correct answer. However, let’s also check why the last two answer choices can’t be correct.

(D) j is divisible by k

Let’s say j = 2 and k = 4: we have 2^2/4 = 1, but j is not divisible by k. So, D is not the answer.

(E) j^2 > k

Let’s say j = 1 and k = 1: we have 1^2/1 = 1, but j^2 is not greater than k. So, E is not the answer.

Answer: C
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Hi...
You can easily find others NOT to be true in all cases..
But D says.. if j is even, k is even...
Say j is odd but k is even j^2/k would not be an integer itself..
So this must be TRUE

D
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The only way to obtain an odd quotient is if either of these two cases occurs: 1) both numerator and denominator are odd, such as 9/3 = 3, or 2) both numerator and denominator are even, such as 6/2 = 3. Secondly, if an integer is odd, its square is also odd, and if an integer is even, its square is even. Since j^2/k is odd, we see that if j itself is odd, then j^2 is odd, and so k must be odd. OR, if j is even, then j^2 is even, and k must be even.

Thus, of the answer choices, only D must be true.

Answer: D
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if j is even, then j^2 is even, and k must be even.
then how would the answer be odd..

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