GMAT Tip of the Week: It's Hip to be Square

For GMAT instructors and number enthusiasts, yesterday was a banner day – on April 25, 4/25, both the month and the day (4 and 25) were perfect squares (2-squared and 5-squared). And with that in mind, let’s take a look at some properties of squares that can help you better solve exponent questions on the GMAT.

1) Squares mean that all prime factors are doubled

The definition of a square is that it’s the same thing times itself. Which means that you have two of everything. Take 6^2 – it’s 6 * 6, and if you broke those 6s down to their prime factors you’d have (2*3) * (2*3), meaning that you have two factors of 2 and two factors of 3. If you’re squaring an integer, that means that each of its prime factors are doubled – those factors must come in pairs.

How can this be helpful? The GMAT tends to ask questions such as: For positive integers x and y, if 75x = y^2, which of the following must be a factor of x?

And in this case, if you prime factor out the only given number, 75, you’ll see that you have: 3*5*5*x = y*y

This means that x MUST contribute a factor of 3 to the pairing, since you need to have two of each factor and right now you only have one 3. So x MUST BE divisible by 3 (but since y could be even bigger than 15, you don’t know that x is exactly 3…the setup could be 2*3*5*2*3*5 = y*y. But you do know that x must have a factor of 3).

When you’re dealing with integers squared, know that the prime factors must then come in pairs.

2) Squares have an odd number of factors.

When thinking in terms of factors of integers, you should recognize that every factor of a number x must multiply by another number to produce x. So factors come in pairs – take 42: Its factors are 1 and 42; 2 and 21; 3 and 14; and 6 and 7. Each factor has a pair with which it multiplies to 42. But squares have an interesting property in that one of their factors doesn’t have a *different* pair – it multiplies by itself to produce the square. Take 36: its factors are 1 and 36; 2 and 18; 3 and 12; 4 and 9; and 6 and…well, 6. So squares break that mold of “all factors come in pairs”, because one of the factors goes solo and multiplies by itself.

Perhaps more unique – squares of prime numbers have exactly three unique factors: Itself, one, and the square root. Take 9 – its factors are 1, 3 and 9. So if a number is designed as having exactly three factors, you know it’s the square of a prime number.

3) To get to the next square, “square it off”

This is a relatively rare property but understanding it can help you unlock many difficult number properties problems. Before you take the GMAT, you should absolutely know the squares from 1-15, and you should know that 25-squared is 625. But much past that the ROI on memorization gets pretty low. But here’s a way to think about larger squares if you do ever need to calculate them. Take 41-squared. You should quite easily know that 40*40 is 1600. How do you get from there to 41*41? Well, 41*40 is going to be 40*40 + 40 – if you already know what forty 40s looks like, forty-one 40s is just one more 40. So that takes you to 1640. And since you now have 41*40, or forty 41s, in order to get to forty-one 41s, you just add one more 41. So you go from 1640 to 1681, and that’s 41-squared.

This property – a quick way to calculate larger squares – derives directly from the term “squared”. If you think of 3*3 visually, it’s three rows of 3:

X X X
X X X
X X X

And if you want to go from 3×3 to 4×4, first you have to add a fourth row:

X X X
X X X
X X X
X X X

But now it’s a 3×4 rectangle, so you need to “square it off”. You’ve added three more already, but now you need to add a column of 4 to square it off:

X X X X
X X X X
X X X X
X X X X

So what you’ve done to get from 3×3 to 4×4 is add 3 (to get to 3×4) and then add 4 (to get to 4×4). So 3^2 (which is 9) is 7 away from 4^2 (which is 16).

Note that the GMAT likes to test exponents, factors/multiples, geometry (squares are big in the Pythagorean Theorem and in the shapes squares themselves) and unique number properties, so squares have plenty of opportunities to come into play on the exam. Better understand squares and you’ll find…it’s hip to be square.

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