My very late 2019 problem
A rectangular lawn with integer dimensions has perimeter and has area . Given that, the rectangle's dimensions give the same remainder when divided by , input the perimeter of the rectangle.
If there isn't any rectangle with those requirements, input .
The answer is 280.
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1 solution
Here's the solution:
Perimeter = 2L+2W = k (equation 1)
Area = LW = k+2019 (equation 2)
Therefore, we can substitute equation 1 into equation 2:
LW = (2L+2W) + 2019
LW -2L -2W = 2019
Add 4 to both sides
LW -2L - 2W +4 = 2023
Factoring the left side gives:
(L-2) (W-2) = 2023
Now, we can use the prime factorization of 2023:
2023 = 7 × 17²
Which means these are the possible pairs of positive numbers which multiply to 2023:
(1, 2023) (7, 289) and (17, 119)
These lead to these possible dimensions: (3, 2025) (9, 291) and (19, 121)
The only group where the dimensions give the same remainder when divided by 17 is when the dimensions are 19 and 121.
Therefore, the perimeter is 2×19+2×121 or 280