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This essential trains for: SAT-I, SAT-II, GMAT, AMC-8, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.

A perfect square is a positive integer in whose prime factorization all the exponents are even.

Thus, it is the result of multiplying some (other) integer by itself.

Example:

N is the square of:

Obviously,

Since, according to the fundamental theorem of arithmetic, any positive integer has a unique prime factorization, it is always possible to tell if a number is a perfect square or not by factoring it into primes.

The last digit of a perfect square: cannot be any digit. Rather, think of what happens when we multiply numbers ending in various digits by themselves - we can come up with the following table:

Last digit of n | Last digit of n2 |

0 | 0 |

1 | 1 |

2 | 4 |

3 | 9 |

4 | 6 |

5 | 5 |

6 | 6 |

7 | 9 |

8 | 4 |

9 | 1 |

We see that perfect squares can never end in 2, 3, 7 or 8.

Consecutive perfect squares are squares of consecutive integers:

and, therefore there is no perfect square between n2 and (n+1)2.

Division by 4: can only yield remainders of 0 or 1. Since the possible remainders upon division by 4 are:

by squaring these equivalences we find the possible remainders of any perfect square:

Any perfect square can be written in a form that matches the algebraic identities for perfect squares:

This observation can be used to help the mental computation of certain expressions like this example:

or this example: