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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.
N is the square of:
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|
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: