MathLabs

Problem 2

Find the largest positive integer NN such that the number of integers in {1,2,…,N}\{1,2,\dots,N\} divisible by 33 equals the number of integers divisible by 55 or 77 (or both).
Step 3 of 6: Obtain the floor equation
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D(N):=⌊N3⌋−⌊N5⌋−⌊N7⌋+⌊N35⌋=0.D(N):=\left\lfloor\frac N3\right\rfloor-\left\lfloor\frac N5\right\rfloor-\left\lfloor\frac N7\right\rfloor+\left\lfloor\frac N{35}\right\rfloor=0.
Detailed analysis

The required integers are exactly the positive N for which D(N) is zero.