# If j and k are integers and j-k is even, which of the following must be

If \$j\$ and \$k\$ are integers and \$j-k\$ is even, which of the following must be even?

1. \$k\$
2. \$jk\$
3. \$j+2k\$
4. \$jk+j\$
5. \$jk-2j\$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 19 of Section 6 of Practice Test 1. Those questions testing our knowledge of Integers can be kind of tricky, but never fear, PrepScholar has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

This question asks us about even and odd integers, which is a clue that it likely tests our Integers math skill. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

1. We know that \$j\$ and \$k\$ are integers
2. We know that \$j-k\$ is even
3. We want to know which answer choice MUST be even

## Develop a Plan

We want to know which answer choice MUST be even. So if we can find a case where an answer choice CAN be odd, then we can cross it off as incorrect. Since this question heavily involves determining if expressions are even and odd, let’s first begin by reviewing even-odd integer arithmetic.

## Concept Refresher – Even-Odd Arithmetic Rules

Even numbers are integers with a units digit of \$0, 2, 4, 6, \or \;8\$, whereas odd numbers are integers with a units digit of \$1, 3, 5, 7, \or \;9\$. The rules for adding even and odd numbers are the same as the rules for subtracting them, so we’ll write them both together:

\$\$\Even ± \Even = \Even\$\$
\$\$\Even ± \Odd = \Odd\$\$
\$\$\;\;\,\Odd ± \Odd = \Even\$\$

So to get an odd number when adding or subtracting two numbers together, we must have exactly one of the numbers be odd. Now let’s review the rules for multiplying even and odd numbers.

\$\$\Even · \Even = \Even\$\$
\$\$\;\,\Even · \Odd = \Even\$\$
\$\$\;\;\Odd · \Odd = \Odd\$\$

So to get an odd number when multiplying two numbers together, we must have both numbers be odd. Now let’s get back to the question at hand.

We know that \$j-k\$ is even. From our even-odd arithmetic rules, we can see that if the difference between two numbers is even, then either: 1) both terms are even or 2) both terms are odd. So our plan will be to check two cases for each answer choice: 1) both \$j\$ and \$k\$ are even, or 2) both \$j\$ and \$k\$ are odd. We can cross off an answer choice if we find that it gives us an odd result for either case.

## Solve the Question

### Check A: \$k\$

We know that \$k\$ could be odd or even, so since it CAN be odd, we can cross off this answer choice.

### Check B: \$jk\$

If both \$j\$ and \$k\$ are odd, then \$jk = \Odd·\Odd = \Odd\$. So B is incorrect.

### Check C: \$j+2k\$

If both \$j\$ and \$k\$ are odd, then we have:

 \$j+2k\$ \$=\$ \$\Odd + \Even·\Odd\$ \$ \$ \$ \$ \$j+2k\$ \$=\$ \$\Odd + \Even\$ \$ \$ \$ \$ \$j+2k\$ \$=\$ \$\Odd\$

Since C can be Odd, then it’s incorrect.

### Check D: \$jk+j\$

If both \$j\$ and \$k\$ are odd, then we have:

 \$jk+j\$ \$=\$ \$(\Odd·\Odd) + \Odd\$ \$jk+j\$ \$=\$ \$\Odd + \Odd\$ \$jk+j\$ \$=\$ \$\Even\$

If both \$j\$ and \$k\$ are even, then we have:

 \$jk+j\$ \$=\$ \$(\Even·\Even) + \Even\$ \$jk+j\$ \$=\$ \$\Even + \Even\$ \$jk+j\$ \$=\$ \$\Even\$

Ah! So in both cases, D cannot be odd. So D is correct. Since we’re having fun with this, we might as well check E.

### Check E: \$jk-2j\$

If both \$j\$ and \$k\$ are odd, then we have:

 \$jk-2j\$ \$=\$ \$(\Odd·\Odd) – (\Even·\Odd\$) \$jk-2j\$ \$=\$ \$\Odd – \Even\$ \$jk-2j\$ \$=\$ \$\Odd\$

Since E can be odd, it’s incorrect.

The correct answer is D, \$jk+j\$.

## What Did We Learn

We should commit to memory the even-odd arithmetic rules for addition, subtraction, and multiplication. Of course, if we ever forget them during the test (we are humans after all, not computers) we could quickly do a math experiment to figure them out. The next best thing to knowing something is knowing how to figure it out! So for example, if we forgot the result when we multiply two odd numbers together, then we could just choose two odd numbers, multiply them together, and see what we get. For example, we could multiply \$3\$ and \$5\$ together to get \$15\$. Since \$15\$ is odd, we know that multiplying two odd numbers gives us an odd number. We should always feel free to do this during the test if we’re not sure that we remembered these rules correctly.

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