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 5 of 6: Check the largest candidate
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D(65)=⌊653⌋−⌊655⌋−⌊657⌋+⌊6535⌋=21−13−9+1=0.D(65)=\left\lfloor\frac{65}3\right\rfloor-\left\lfloor\frac{65}5\right\rfloor-\left\lfloor\frac{65}7\right\rfloor+\left\lfloor\frac{65}{35}\right\rfloor=21-13-9+1=0.
Detailed analysis

At N=65 the two counts are both 21, so 65 satisfies the condition.