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 1 of 6: Translate the first count
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#{m≤N:3∣m}=⌊N3⌋.\#\{m\le N:3\mid m\}=\left\lfloor\frac N3\right\rfloor.
Detailed analysis

Among the positive integers up to N, exactly the multiples of 3 are counted, so the count is the floor of N divided by 3.