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Squares of Integers
Squares of integers, also known as perfect squares. Here are the first 15 perfect squares, absolutely beautiful numbers. Certainly know the first ten. Those first ten, you just have to know inside-out, forward and backwards, you have to know them cold. Knowing and recognizing the last five will save you time on the harder math problems, so it’s really a good idea to know this whole list.
Let’s think about one square a little more, for example, 12 squared equals 144. Let’s find the prime factorization of this. Well of course, 144 equals 12 times 12. Each factor of 12 is 2 times 2 times 3. And then when we combine, we get 2 to the 4th, 3 squared. Now what’s interesting here, of course, each factor was, as it were, doubled up because it appeared once in each factor of 12.
So 3 appears once in 12, and so it has to be here twice in 144. 2 appears twice in 12, so it has to be here 4 times, in 144. So, both the exponents in 144 had to be even. And in fact, this is generally the case. The exponents of the prime factors of squares all must be even. You always get this doubling up, any factor that appears once in N is gonna appear twice in N squared.
If it appears twice in N, it’s gonna appear 4 times in N squared, that sort of thing. This means that if we see an unknown number in its prime factorization form, and all the exponents are even, we know right away the number must be a perfect square.
For example, suppose we’re given this number, K equals 2 to the 6th, times 3 to the 4th, times 5 squared. Well,we don’t know what that number is, it’s probably a big number. But we can tell right away just by looking at the prime factorization, that is definitely a perfect square because 6, 4, 2, all those exponents are even numbers.
Right away we can tell that. And in fact, we can go a little bit further here. We know that K must be some factor times itself, and that factor would be 2 to the 3rd, 3 squared, times 5. Notice all the exponents divided by 2, because they’re gonna doubled up when we multiply them together.
And in fact, that number in parenthesis, we can figure out very easily. That number in parenthesis, 2 cubed times 3 squared, times 5, 8 times 9 times 5, that’s 40 times 5, 40 times 9, which is 360. So, 360 would be the square root of K, and K would be 360 squared. So, all of this we can figure out very easily from the prime factorization. Now counting factors in a perfect square.
In the last video, we talked about a procedure for counting the number of factors in a large number. If you haven’t seen that video, you really need to go and watch that video, so the video before this one and that way what I’m gonna say here will make sense. It will make sense when I’m talking about the lists, if you don’t, if you’re not, you’re not familiar with the procedures in the last video.
So, let’s pretend that we’re gonna count the factors in a perfect square. Notice, since all the powers of the prime factors of a perfect square are even, when we’re counting the factors, the list of exponents will be a list of all even numbers. Then when we add 1 to each number on the list to get the second list, the second list will be a list of all odd numbers.
Because of course when you add 1 to an even number you get an odd number. This means that the product of that second list will be a product of all odd numbers, so it will have to be odd. And, of course, the product of that second list is the number of factors, therefore, a perfect square always has an odd number of factors. That’s a really big idea.
Another way to see this is to think about factor pairs. Think about the factors of the perfect square 36. Think about the factor pairs. Well, of course one pair is 1 times 36. Of course, 1 if a factor of every number, every number is a factor of itself. 2 times 18, 3 times 12, 4 times 9, all these factors in pairs.
But then we get down to 6 times 6, and of course, that counts as only one factor 6. Well, this is interesting. Every other factor was in pairs but 6, which is the square root of 36, is all by itself. And thus, we have an odd number of factors because we have one unpaired factor. And of course this is only gonna happen for perfect squares.
If a number is not a perfect square, all the factors are gonna be in pairs. The only time you get an unpaired factor is when you have a perfect square. So on, one factor of N squared will always be N, which counts as only one factor because it’s in a pair with itself, and perfect squares are the only integers with an odd number of factors.
So, in summary, these are very important ideas for advanced questions on integer properties. In the prime factorization of a perfect square, each prime factor must have an even exponent. If the prime factorization of some unknown integer has all even exponents, that unknown integer must be a perfect square.
Perfect squares always have an odd number of factors. And the only integers with odd numbers of factors are the perfect squares.
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