Solving a nice floor equation

So today we will be solving a nice floor equation . these type of problem are really nice to solve and satisfying .

So if you have any other method other than mine or else if you have any other question that you think I should try you can Email me or else you can send it to me on my Instagram .

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A huge thanks to Vincent Rubinetti for the background Music

#math algebra #mathematics

[elihowitt]

Inspired by the change of thumbnail after I generalized for any RHS motivated me to generalize for any similar problem (this would be a tool to prove insolvability):

For any integers a, b, c, d; consider the function f(x) = floor(ax + b) + floor(cx + d). Let e = (a+c)/gcd(a, c).
f(x) is incongruent to b + d - 1 mod e.

This is useful for example where originally the thumbnail had LHS = floor(x+1) + floor(2x+1), (a=1, b=1, c=2, d=1) and RHS = 13-
f(x) = LHS is incongruent to {b+d-1 = 1+1-1 = 1} mod {e = (a+c)/gcd(a, c) = 3/1 = 3}, hoever RHS = 13 is congruent to 1 mod 3 => no solution!

Another example: floor(4x+3) + floor(6x+5).
a = 4, b = 3, c = 6, d = 5.
f(x) is therefore incongruent to 2 mod 5 => f(x) doesn't map to {..., -8, -3, 2, 7, 12, 17, ...} .

Pretty cool! message me if you're interested in the proof or if you can generalize further (e.g to any rational a & c, or for >2 floors (seems simple but-)) I couldn't be bothered atm.

[MyOneFiftiethOfADollar]

I also got [3.5, 4) by a different method. If you drop the floor notation, you get 3x+3 = 13 or x = 10/3 where we can conclude the solution will be an interval "near" 10/3 for floor(x+1) + floor(2x+2) = 13.

Also, it helps to consider where the linear equation y=3x+3 would intersect horizontal line y = 13 to gain insight on the location of solution interval.

Also I tried the method on floor(x+1) + floor(2x+2) = 14 and got no solution. Thanks for problem.

[yoav]

Oh this is +2, this was 2x+1 at the begining and i wonder how is there asolution🤣.but nice problem well done!!

[nirmankhan]

I have a great question on floor bracket. Shall I send it to you in insta??