There has been a huge surge in the number of
questions about perfect squares, in almost all the exams. The
basic trick to any such question is assuming the number as a
perfect square of an integer k and then using techniques of
completion of square and then the formula of and
solving using divisibility theory
Example I: Find all natural n such that
is a perfect square
Step 1:
Step 2:
Step 3:
see now and
both are integers then
both of
and
are divisors
of 64. But note that we add the two equations we will get
so the sum of two divisors should be even hence both
divisors even or both odd
so and
but see this n is positive hence k is positive, thus
so only two options
and solving we get
so
Note : The source of this
problem is Pomona Wisconsin mathematics talent search exam!
Practice problem!!
Find the sum of all such positive integers m's such that
is a perfect square
Now we will extend the method to other kinds of
problems
Basically what we used in the above problem is difference of
square method
lets take an example
find
the no of pairs of positive integers
first step in this problem is recognizing that 127 is a prime
then we move to
so clearly
x = 4 and y = 63
so one pair (4,63)
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