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a = 4

b = 7

Find values of c. Such that a*c-3 and b*c-3 both are perfect square. 

I have already found 2 answers 1 and 21 I need to find more.

Thanks.

 Oct 1, 2022
 #1
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ac - 3 and bc - 3 are perfect squares. (also i am assuming that c is an integer cuz duh)

4c - 3 and 7c - 3 are perfect squares

4c - 3 = p^2, 7c - 3 = q^2

Let's do the 4c - 3 first: p^2 must be odd because when you add 3, it gets even and only evens can be divided by 4.

p^2 + 3 is divisible by 4.

p^2 must be odd, so the squared can be 1, 3, 5, 7 (note how all these +3 are divisible by 4), 9, etc.

So 4c - 3 = p^2 doesn't really help. All we can conclude is that c >= 1 and is an integer.

7c - 3 = q^2: q^2 + 3 is divisible by 7. Yowch, divisiblity of 7 is annoying... q = 2 works, 5 works, 9 works, 12 works, ...

 

To find more solutions, just make sure that q^2 + 3 is divisible by 7, find the possible values of q, and then use algebra to deduce c... There are a bunch of different solutions (so long as c >= 1 and is an integer).

 Oct 2, 2022

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