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Algebra Level 3

The number of students in a University is a perfect square.In one year 2000 more students joined the original strength and the new strength is one more than a perfect square. The next year 2000 more students joined newly and the new strength of the university is again a perfect square.

Then what was the original strength of the university?


The answer is 998001.

Naitik Sanghavi by naitik sanghavi , 20, India

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1 solution 

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Let k**2 = original number of students

Then:
k**2 + 2000 = 1 + m**2
OR
k**2 + 1999 = m**2 <Equation 1>

k**2 + 2000 + 2000 = p**2
OR
k**2 + 4000 = p**2 <Equation 2>

<Equation 2> - <Equation 1> :
p**2 - m**2 = 2001
OR
(p+m)(p-m) = 2001(1)

p + m = 2001
p - m  = 1

p = 1001
m = 1000

There are other integral values for p and m but those values 
will only make k**2 negative

From <Equation 1>:
k**2 + 1999 = m**2
k**2 + 1999 = 1000**2
k**2 = 998001 <Perfect square>

Answer: 998001

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