Discussion
Find the number of integers between 1 and 10,000 both inclusive which are divisible by none of 5, 6 or 8 :
Answer: D
The number is 6000.
Use inclusion-exclusion on 1 to 10,000. Multiples of 5, 6 and 8: \(\lfloor 10000/5 \rfloor = 2000\), \(\lfloor 10000/6 \rfloor = 1666\), \(\lfloor 10000/8 \rfloor = 1250\). Pairwise, with lcm(5, 6) = 30, lcm(5, 8) = 40 and lcm(6, 8) = 24: 333, 250 and 416. All three, lcm 120: 83.
\[ \begin{aligned} |A \cup B \cup C| &= 4916 - 999 + 83 \\ &= 4000 \end{aligned} \]
So \(10000 - 4000 = 6000\) are divisible by none of them. The trap is the pair 6 and 8, whose lcm is 24, not 48; using 48 changes the count.
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