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Sunita grows potatoes in his backyard which is in the shape of a square. Each potato takes 1 cm2 in his backyard. This year, she has been able to grow 133 more potatoes than last year. The shape of the backyard remained a square. How many potatoes did Sunita produce this year?

My attempt:

I have pictured it like this:

enter image description here

If the number of potatoes that was growm previously = $x*x=x^2$

Then the additional potatoes grown this year

=The horizontal and vertical rectangles that are getting added$(x+x=2x)$+the little squares that are getting added as 1+3+5+7+....(odd numbers)

= $(x+x)*k+\sum_{n=0}^k(2n+1)=133 $

Soumee
  • 1,087

1 Answers1

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For $x,y$ the old and new squares, you're told $$x^2+133=y^2$$ or equivalently, $$133=y^2-x^2=(y-x)(y+x)$$

133 has factors $(1,133)$ and $(7,19)$ making it semi-prime.

Now, the part that solves such systems:

If adding $x$ gets to one point but subtracting $x$ gets to a second point, $x$ must have magnitude half the distance between the points. That forces $y$ naturally to be halfway between the two points.

Given this intuition, the solutions are then

$(x,y)$ is either $(66,67)$ or $(6,13)$. This year Sunita grew either 169 or 4489 potatoes.

Nij
  • 2,991