In the second part of our article on Perfect Squares and Cubes, today, let's take a look at the different models of questions that GMAT creates, based on Perfect squares and cubes.
If you missed out on reading the first part of this article, here's the link for you:
Perfect Squares & Cubes - Part 1You can go through the first part and come back and continue with this part from there onwards!
Perfect Squares in Number Properties questionsA very conspicuous feature of any perfect square is that it consists of at least one pair of prime factors. When a perfect square is prime factorized, the powers of the prime factors will always be even numbers.
The exception to this is of course, 1 (as in a lot of cases) Let’s take some simple examples to understand.
4 = \(2^2\) = 2*2; so 4 has a pair of 2s.
16 = \(2^4\) = 2 * 2 * 2 * 2; so 16 has TWO pairs of 2s.
36 = \(2^2\) * \(3^2\) = 2*2 * 3*3; 36 has ONE pair of 2s and ONE pair of 3s.
Corollary – Any positive integer greater than 1, that does not have pairs of prime factors cannot be a perfect square. This property has also been tested in GMAT Quant questions where you could be asked if a given number is a perfect square OR how to convert a given number to a perfect square.
We could take the case of 120, for instance. 120 can be prime factorised to give us,
120 = \(2^3\) * \(3^1\) * \(5^1\) = 2 * 2 * 2 * 3 * 5
We see that, there is only ONE pair of 2, but the other 2 is unpaired and so are 3 and 5.
So, if 120 has to be converted to a Perfect square,
it can either be multiplied with a 2, 3 and 5 each to obtain 3600
or
divided by the product of 2*3*5 to obtain 4.
Both 3600 and 4 are perfect squares.
Number of Factors of a Perfect Square A perfect square always has an ODD number of total factors - which is essentially a result of what we saw in the previous section. You ask how?
For a
composite number that can be written in the form of
N = \(a^p * b^q * c^r\) * …, where
a, b, c etc., are prime factors of N and
p, q, r etc., are positive integers Number of factors = (p+1) (q+1) (r+1)….. For example, if the number N is 24, prime factorizing 24 yields
24 = \(2^3 * 3^1\).
Comparing with the standard form, a = 2 and b = 3, which are the two prime factors of 24; p = 3 and q = 1 which are the respective powers of the prime factors.
Therefore, number of factors of 24 = (3+1) * (1+1) = 4 * 2 = 8. We all know that this is correct because 24 does have 8 factors viz., 1,2,3,4,6,8,12 and 24.
When it comes to perfect squares, in the previous section, we learned that the powers of the prime factors are always even numbers. That is,
if N is a perfect square, then
the values for p, q, r etc., in the prime factorization of N
will be even numbers.
Therefore, Number of factors for
N = (even + 1) (even + 1) (even + 1)…. = odd * odd * odd*….It follows that all the terms in the formula shown above will be odd and hence the final product (which represents the total number of factors of the given number) will be odd.
Taking the example of 4 = \(2^2\), number of factors of 4 = (2+1) = 3. This is correct – 4 does have exactly 3 factors namely 1, 2 and 4.
Similarly, 36 = \(2^2 * 3^2\); number of factors for 36 = (2+1) * (2+1) = 3 * 3 = 9.
Therefore,
the total number of factors will always be ODD for a Perfect square. What is great about this feature of perfect squares is the fact that the vice-versa is true as well.
This means –
Any number that is found to have an odd number of factors HAS to be a perfect square. This is a property that has been tested quite often in questions related to Factors and Multiples. So, on a GMAT question, if you come across a statement like “\(x\) has an odd number of factors”, GMAT is trying to tell you that “\(x\) is a perfect square”.
Perfect Squares in Algebra In Algebra, perfect squares are usually represented as exponents having even powers viz., \(x^2\), \(x^4\) and so on.
In a reiteration of a property that we discussed in our article on Zero, One and Two, this is to reiterate that
the smallest value for such even powered exponents, is Zero. (Curious to read our article on Zero One Two? Click on this link -
Zero One Two )
One of the most common bad habits we carry from our school days is to remember that
“\(x^2\) is always positive”. If you really want to avoid traps in Algebra questions on the GMAT, this is one of the
habits to give up and replace it with
“\(x^2\) is always non-negative”.
When you do this, you will go into the analysis of an expression / equation with a more open mind, knowing that \(x\) could be zero; you will also look consciously for the possibility of \(x≠0\) being given as data. If this is given, you can now conclude that \(x^2\) MUST be positive. This two-step process will really change the way you analyze and synthesize algebraic expressions and equations on the GMAT.
Yet another trap laid out for the unsuspecting test taker is in Data Sufficiency questions, where a number like \(x^2\) is given to be an integer. A very common mistake
that students make here is
to conclude that \(x\) is also an integer because \(x^2\) is an integer. This is not correct. If \(x^2\) is an integer, \(x\) could be an integer or a root. For example, if \(x^2\) = 4, x = +2 or –2, both of which are integers; however, if \(x^2\) = 5, x = \(\sqrt{5}\) which is not an integer.
Corollary – Look out for a statement which says “\(x^2\) is a perfect square”; only in this case can you conclude that \(x\) is an integer because a perfect square is the square of an integer.
Another area where your knowledge of Perfect squares can help is when you must approximate the value of roots.
For example, if the square root of 8 must be calculated, the typical approach is to say that 8 = 4 * 2, therefore, the square root is 2*\(\sqrt{2}\) and simplify it to 2.8 since \(\sqrt{2}\) = 1.414.
However, there’s a faster method than this, which flips the entire process of finding roots on its head and deals with squares instead.
Now, here’s how you can utilize your knowledge of squares. Instead of 8, let’s try asking ourselves “what’s the square root of 800?”
From the table, we see that \(28^2\) = 784 and \(29^2\) = 841. Clearly, the square root of 800 has to be closer to 28 since 800 is closer to 784. Therefore, 28 is a better approximate for the square root of 800.
But 800 = 8 * 100. Therefore, \(\sqrt{800}\)=\( \sqrt{8∗100}\).
Since \(\sqrt{800}\)=28 and \(\sqrt{100}\)=10,
28 = \(\sqrt{8}\) * 10 which gives us
\(\sqrt{8}\) = 28 / 10 = 2.8
Now that you have understood this process, can you find the approximate square root of 38?
Perfect Squares in Sequences In questions on sequences and progressions, there are two situations where your knowledge of perfect squares can be tested.
The sum of the first ‘n’ perfect squares i.e., \(1^2 + 2^2 + 3^2 + 4^2 + ……. + n^2\), can be calculated using the expression \(\frac{n(n+1) (2n+1) }{ 6}\) 1) There have been questions in which the sum of a certain number of perfect squares has to be calculated, where this formula can be applied directly.
2) It can also be applied in questions where one needs to find the number of squares in a grid.
For example, if we have a 5 x 5 grid of unit squares and we need to calculate the total number of squares, it’s easy to see that the total number of squares is given by the expression \(1^2 + 2^2 + 3^2 + 4^2 + 5^2\) whose final value can be calculated by using the expression shown above.
Another useful expression for solving questions on sequences, related to perfect squares is the sum of a set of odd integers.
The sum of the first ‘n’ odd positive integers is given by \(n^2\). For example, the sum of the sequence 1 + 3 + 5 + 7 + … till 51 will be \(26^2\) since we are taking the first 26 odd positive integers. As we know, \(26^2\) = 676 and this becomes a 30 second answer.
Perfect Squares in Geometry We started off by inferring that the study of Perfect squares probably had its origins in Geometry. Therefore, it wouldn’t be too far-fetched to say that Perfect squares do find application in certain areas of Geometry.
1) Quite obviously, questions involving finding the area / perimeter of a square are natural arenas for you to be tested on your knowledge of Perfect squares.
2) Another topic where your knowledge of perfect squares can come in handy is in questions on Right Angled triangles.
A right-angled triangle is a triangle that follows the Pythagorean theorem, which is
based on the squares of the three sides of the triangle.
Positive integer values that satisfy the Pythagorean theorem are known as Pythagorean triplets / Pythagorean triples. When you know your perfect squares by memory, it becomes easy to recognize Pythagorean triples / missing components of Pythagorean triples.
The Pythagorean triples seen most often in questions are:
3,4,5 and its multiples like (6,8,10), (9,12,15), (12,16,20), (15,20,25) and so on.
5,12,13 and its multiples
8,15,17 and its multiples
7,24,25 and its multiples
20,21,29 and its multiples (Watch out for an interesting article on Pythagorean triples, coming soon )By now, it's clear that the various properties of Perfect squares are tested in different sections of Quant. So, please make sure that you are clear with all the properties we discussed in this article about Perfect squares, so that you can maximize your returns in such questions.
Well, that brings us to the end of our discussion on the properties of Perfect squares. In the third and last part of this article, tomorrow, we will be doing a similar discussion of the properties of Perfect Cubes, stay locked to this forum for that!
Ciao!