Welcome back to our series on number properties. In the last article, we explored the factors of perfect squares. This article will examine a special property of consecutive perfect squares.

A commonly tested skill on the GMAT is the ability to recognize and manipulate an algebraic expression known as a difference of squares. Such an expression represents the difference (in terms of subtraction) between two square numbers.

x2 – y2

x and y need not be integers in such an expression, but they usually are. GMAT problems typically require the following algebraic manipulation of the expression:

x2 – y2 = (x + y)(x – y)

A full treatment of the difference of squares expression and its applications belongs more properly to a series on algebra. For number properties, we will look only at the special case of consecutive perfect squares.

Let’s begin by observing a pattern created by the first few perfect squares:

12 = 1

22 = 4

32 = 9

42 = 16

52 = 25

What are the differences between each two consecutive perfect squares?

(1 – 0 = 1)

4 – 1 = 3

9 – 4 = 5

19 – 9 = 7

25 – 16 = 9

We can see that the differences between consecutive perfect squares follow the series of consecutive odd numbers. Here’s a visual representation that reveals the reason for this pattern:

Squares numbers are called square numbers for a reason: each one can be represented spatially as a square, where the area of the square is the square number and the length of the square’s side is the square root. This is one of the most fundamental geometry and area concepts.

A look at the expanding blue square above reveals the reason behind the “consecutive odd numbers” pattern from each perfect square to the next. In order to produce the next square, expansion is necessary not only in the rightward and downward directions, but also in the “southeast” direction (the numbered squares forming the diagonal line).

From this observation, the following algebraic representation is possible:

x2 – y2 = 2y + 1

Since 2y will always be even (it is some integer multiplied by 2), 2y + 1 (the difference between the consecutive squares) will always be odd. Here’s an example. You can use the visual above to connect the math to the spatial reality.

32 – 22 = 2(2) + 1

9 – 4 = 4 + 1

9 – 4 = 5

So x2 – y2 = 2y + 1. Let’s adjust our “language” on the right side of this equation:

x2 – y2 = 2y + 1

x2 – y2 = y + y + 1

Since x2 and y2 are consecutive perfect squares, x and y are consecutive integers, which means that x = y + 1.

x2 – y2 = y + (y + 1)

x2 – y2 = y + x

To verbalize this algebraic discovery: the difference between any two consecutive perfect squares is equal to the sum of their square roots. Here are some examples:

52 – 42 = 4 + 5

25 – 16 = 9

92 – 82 = 8 + 9

81 – 64 = 17

It works! But if the general difference of squares formula (the one we saw at the top of the article) looks like this:

x2 – y2 = (x + y)(x – y)

then what makes consecutive squares special? What happened to the (x – y) term? Well, since for any pair of consecutive squares, x = y + 1, then x – y = 1. In other words, the formula for the difference between consecutive squares is not really a special case – it’s simply the case where the (x – y) term equals 1 and thus “disappears” since it has no effect on the value of (x + y)(x – y).

For consecutive perfect squares:

x2 – y2 = (x + y)(x – y)

(x – y) = 1

x2 – y2 = (x + y)(1)

x2 – y2 = (x + y)

Let’s put this discovery to work on an official GMAT data sufficiency problem:

If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

  1. 69 + n and 94 + n are the squares of two consecutive integers.
  2. 94 + n and 121 + n are the squares of two consecutive integers.

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are not sufficient.

Each statement gives us expressions for “the squares of two consecutive integers” – in other words, two consecutive squares. Since these statements are qualitatively similar, it is very likely that the correct answer will be D or E: either this type of information is sufficient in each case (answer D), or it isn’t sufficient at all (answer E). Let’s work with statement 1.

x2 – y2 = x + y

x2 = (94 + n)

y2 = (69 + n)

(94 + n) – (69 + n) = x + y

25 = x + y

To return to our earlier verbalization, the difference (25) between these consecutive squares is equal to the sum of their square roots, which are consecutive integers. So x and y are consecutive integers that add up to 25. The only solution is 12 and 13. We can find these values by considering that 25/2 = 12.5 and taking the “adjacent” integers of 12 and 13.

25 = 13 + 12

x = 13

y = 12

If x2 = (94 + n) and y2 = (69 + n), then the solutions for the values of x and y mean that the only variable remaining is n. Therefore we can solve for n, and statement 1 is sufficient.

x2 = 94 + n

x = 13

132 = 94 + n

169 = 94 + n

n = 75

Now looking back at statement 2, we see that it provides expressions for the next two consecutive squares, since it again uses (94 + n) as the expression for one of these squares. Therefore the expressions (94 + n) and (121 + n), respectively, represent 132 and 142, and statement 2 by itself could allow us to solve for n just as statement 1 did.

121 + n = 142

121 + n = 196

n = 75

Therefore the correct answer is D: each statement alone is sufficient.

This concludes our study of consecutive perfect squares. In a nutshell, for consecutive perfect squares, x2 – y2 = x + y, or to put it into words, the difference between consecutive perfect squares is equal to the sum of their square roots. In the next article, we will return to the heart of number properties: divisibility and its relationship to prime factors.

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Contributor: Elijah Mize (Apex GMAT Instructor)

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