Square numbers are numbers which have all of their prime factors raised to even powers. For example if we take the number 36, the prime factors of 36 are 2 and 3 but both of these are raised to the power of 2 in other words 2 squared and 3 squared. If we multiply 2 squared and 3 squared together we arrive at 36:


In a similar example we can see that the number 256 is also a square number because it is 2 raised to the power of 8:


Let us take a look at some examples of using square numbers:


Q. Write out these numbers as products of their prime factors and then find the smallest integer that you could multiply each one by to give the square number..... 180, 250 and 416


A1. 180, divide by 2 to give us 90, divide again by 2 to give us 45, divide by 3 to give us 15 and divide again by 3 to give us 5. Therefore the prime factored representation of 180 is 22 x 32 x 5. to make a square number, where each prime factor is raised to an even power we would need to multiply by 5. As things stand, we have 2 raised to the power of 2, multiplied by 3 raised to the power of 2, multiplied by 5 raised to the power of 1. We multiply once again by 5 to make it 5 to the power of 2 and we end up with 900 which is a square number.


So the smallest number that we can multiply by in this instance is 5.


A2. 250, divide by 2 to give us 125, then divide by 5 to give us 25 and divide again by 5 to give us 5. This means that 250 in its prime factorised state would be 21 x 53 = 250. The smallest number that we could multiply this by to make it a square number would have to be a number with prime factors that would increase the powers we have sufficiently to make them both even, in other words 21 would need to be multiplied again by 2 to the power of 1 and then 5 to the power of 3 would need to be multiplied by 5 to the power of 1 to make 5 to the power of 4. 


(21 x 53) x (21 x 51) = 250 x 10 = 2500. 


It is only the second part that we are interested in, that is (21 x 51) = 10


So the answer to the question is that we need to multiply 250 x 10 to make it a square number.


A3. 416, divide by 2 to make it 208 and divide again by 2 to make it 104, divide again by 2 to make it 52 and then again by 2 to make it 26. Divide again once more to make it 13 which is where the story ends because 13 is a prime number. Therefore 416 is 25 x 131 = 416.



In case you're struggling to see this, let us look at the penultimate part of the answer to the previous question. We have an odd power (5) and another on power (1). These need to be made into even powers so that our statement would actually be 26 x 132 = 10816. to make 25 into 26 we multiplied by 21  and to make 131 into 132 we multiplied by 131 . what we did effectively is multiply the whole of the first expression (the one which gave us 416) by 21 x 13 = 26 and this is in fact the number that we want, so to make 416 into a square number we have to multiply by 26.


Again, you may be given a question in a "story -like" format and your first task in the circumstances will be to actually work out what the question wants you to give us an answer, in other words you have to understand the question and effectively rewrite it mathematically before you attempt to solve it. Let us now take a look at a couple of examples of this:


>> Questions <<