# Solving problems with perfect squares in GMAT Quant

An important area of GMAT math is the concept of a perfect square. First, we should define it. A perfect square is an integer that is the square of an integer. Let’s look at an example of a question involving perfect squares from GMAT Tutor’s bank of practice questions:

*x and y are integers. If the units digit of x ^{2} is 9, and the units digit of y^{2} is 9 as well, which of the following CANNOT be (x−y)?*

*A) -4B) 4C) 6D) 10E) 18*

What does the fact that the units digit of both x^{2} and y^{2} is 9 tell us about x and y? Consider the following table:

x | x^{2} |

1 | 1 |

2 | 4 |

3 | 9 |

4 | 16 |

5 | 25 |

6 | 36 |

7 | 49 |

8 | 64 |

9 | 81 |

10 | 100 |

11 | 121 |

12 | 144 |

13 | 169 |

We can use this table to observe that for any integer that ends in 4, for example, its square will end in 6. How does this help us with this question? Note that if x^{2} ends in 9, x must end in 3 or 7. Let’s set up another table to eliminate some of the answer choices. (Note that the underlying value of x or y does not have to be positive — it can also be negative.)

x | y | x – y | |

7 | 3 | 4 | Eliminate B |

3 | 7 | -4 | Eliminate A |

7 | -3 | 10 | Eliminate D |

13 | 7 | 6 | Eliminate C |

E is therefore the correct answer. There is no way to get to a result of 18 for x – y, if both x and y end in either 3 or 7.

You may not have seen this from the beginning, but the process of elimination gets rid of all the other answer choices.

Notice the numbers we chose for the table. The obvious first choice is to use positive numbers for x and y and make x bigger, so we have a positive result for x - y. Our second choice was just to reverse the order of x and y. Thirdly, we introduced the idea that either x or y, or both, could be negative. Fourthly, it’s possible that x or y or both are not single digit integers.

In summary, the process of elimination helped us greatly. The second useful concept was the idea that the last digit of a perfect square tells us something about the last digit of the square root of the perfect square.