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Even and Odd Integers
Even and odd integers. The test absolutely loves to ask about even and odd integers. Do we know that k is even? Do we know that k is odd? This is a large category question, so this video and the following video talk about these questions.
Let’s start at the beginning here. These are the odd numbers. They go on forever in the positive and negative direction. These are the even numbers. They go on forever in the positive and negative direction. So notice a few things.
First of all, zero is an even number. This is very tricky. Zero is not positive, it’s not negative, but it is an even integer. That’s very important and the test likes to ask about that. Evens and odds include both positive and negative numbers. So, if you’re told that x is even or y is odd, you cannot assume that they are automatically positive.
They could be negative. But even and odd pertains only to integers. Any non-integer is neither even nor odd. So if you are told x is even or y is odd, you absolutely know it’s guaranteed those numbers are integers. Very important.
All even numbers are divisible by 2. Even numbers can be expressed as 2k, where k is any integer. The prime factorization of a positive even integer greater than 2 absolutely has to include 2. Now, that’s a very important idea. The prime factorization of an even number always contains 2, maybe a power of 2, but it definitely contains 2 as one of the prime factors.
No odd number is divisible by 2. Odd numbers can be expressed in the form 2k plus 1 or 2k minus 1, where k is any integer. The prime factorization of a positive odd number will never contain a factor of 2. All right. Well, this is interesting.
So 2k plus 1 and 2k minus 1, those algebraic expressions. So as long as we know that k is an integer, those are the algebraic expressions for the general form of an odd number. That’s important to recognize. It’s also important to recognize that if we do the prime factorization of any odd number, we’re guaranteed there will be no factor of 2 in that prime factorization.
The test loves asking about evens and odds and the basic arithmetic operations, add, subtract, multiply, and divide. So let’s get into this. Adding and subtracting evens and odds. Well, first of all, even plus even, or even minus even, that’s definitely even. If we do odd plus odd or odd minus odd, that’s also even.
It’s only when we mix and match. If we do even plus odd, or we do even minus odd or odd minus even, the mixing and matching results come out odd. So we can summarize all this by saying if we add or subtract likes, we get even. So as long as we are dealing with the same thing, two evens or two odds, then we add or subtract and we get an even.
It’s only when we mix and match, when we add or subtract unlikes, that we get an odd. For multiplying, we also have clear rules that are a bit different. Even times an even always gives us an even number. Odd times an odd always gives us an odd number.
Here, if we mix and match, even times odd, we always get an even number. When you think about this, the deeper reasoning is the prime factors of the product have to come from the prime factors of the two numbers we multiplied together. Well, if we’re doing an even times an odd, that even number has the prime factor 2 as part of its prime factorization.
Well, that’s going to be included in the product. The product will also have a prime factor of 2, which automatically makes the product an even number. If we think about odd times odd, well, again, there are no part, factors of 2 in either odd number. And if there are no factors of 2 in the two numbers we multiplied together, that factor of 2 is not gonna magically appear somehow.
It means that we’re gonna wind up with a product that has no factors of 2. And that’s exactly why odd times odd has to be odd. We can summarize and say as long as there’s at least one even factor in a product, the product will be even. The only way a product can be odd is if every single factor is an odd number. That has important implications when we start multiplying more than one number.
For example, suppose we multiply five integers and get an even number. Then, any individual integer could be even or odd. All we know is that at least one of the five factors is even. So, if someone asks us a spec, a specific question, what do we know about c? Well, gosh, anything could be true about C. We, we’re told that is an integer, but it could be even or odd.
We really don’t know. We know that somewhere in there, there is at least one even number hidden. It might be that one is even, might be that twos are even, might be all five are even. We don’t know. There’s very little that we know from the fact that the product is even.
By contrast, if we multiply five integers and get an odd number, well, then we absolutely know, we’re guaranteed each one of those five integers is an odd number. We know for a fact A is an odd number, B is an odd number, etc. We can draw a clear and unambiguous conclusion about each one of the five numbers when the product is odd. Now division.
Notice there are no general rules for evens and odds with division, largely because so often, integer divided by integer does not even equal another integer at all. Largely, you pick two random integers and divide them, you’re going to get a fraction or a decimal. You’re not going to get a whole number.
When the quotient of two integers does happen to be another integer, when that does happen, whether that quotient integer is even or odd depends on what factors cancel and what factors remain. It cannot be predicted simply by whether the two numbers divided were even or odd. So there is some general patterns, but we can’t make predictions. Here are some general patterns.
If we divide an even number by an even number, the result could be even or odd or not an integer at all. If we divide an odd by an odd, this could be an odd number or not an integer at all. If we divide an even by an odd, it could be an even number or not an integer at all. And if we divide an odd by an even, that’s never gonna be an integer place, because there’s always gonna be a factor of 2 in the denominator that has no way of cancelling with anything in the numerator.
And so, that’s just never gonna be an integer. Here’s a practice question. I suggest pausing the video and working this question out. So in this question, we are told P, Q, R, and S are integers. If P is even, and P times Q plus R times S is an odd integer, then which of the following must be true?
Well, let’s think about this for a second. P is even. That automatically makes P times Q an even number. Well, if the sum is odd, the only way we’re gonna get odd is even plus odd. So that means that R times S has to be odd. Well, we know both R and S are integers.
The only way we’re gonna multiply two integers together and get an odd number is if each one is odd, so we know for a fact R is odd and S is odd. We can draw no conclusion about Q because since P is even, it doesn’t really matter what Q is. P times Q is gonna be an even number automatically, simply because P is even. It doesn’t matter if Q is even or odd.
We can draw no conclusion about it. So the only ones that must be true are two and three, and the answer is D. Here’s another practice question. Pause the video and work on this. If P is an odd integer and P squared plus QR is an even integer, which of the following must be true?
Well, first of all, P is an odd integer. Think about this. P squared, P squared, that’s just P times P, odd times odd. So the square of an odd integer is also odd. So, P squared is odd and the only way that the sum is gonna be an even integer if it’s Q times R is an odd integer.
So we have that much. We know that P squared is odd and Q times R must be odd. Now you may tempt, be tempted here to say, okay, that must mean Q and R are both odd integers. But let’s think about this. We’re not guaranteed that Q and R are integers at all.
It could be that Q and R are even, are both odd integers. But it could be also that Q is, for example, an even number, and R is a fraction. Or it could be that both of them are fractions. For example, with those fractions 7 over 3 and 33 over 7, if we multiply those, we get the odd number 11.
So those are two fractions. Their product is an odd integer, but they’re not integers at all. So, in fact, nothing can be concluded because we are not guaranteed that Q and R are integers. Always be very careful about assuming that numbers are integers. You can’t assume that.
You have to be very careful with that. In summary, we talked about a few ways of thinking about even and odd numbers. We talked about the rules for adding and subtracting even and odd numbers. We talked about the rules for multiplying. We’ve said that there are no general rules for division, although we did talk about some of the patterns.
And then most important, beware of assuming that numbers are integers when that is not stated explicitly. That is one of the major strategies for any number property problems on the test.
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