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 6 of 6: Exclude the remaining larger values
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D(66),D(67),D(68)>0.D(66),D(67),D(68)>0.
Detailed analysis

Direct substitution gives positive values for 66, 67, and 68; together with N<69, no larger solution remains. Therefore the answer is N=65.