<p>How many integers between 100 and 1500 (both inclusive) are multiples of 5 or 11?</p>
Step-by-Step Solution
Key Concept: Inclusion–Exclusion: |M5 \cup M11| = |M5| + |M11| - |M55|. Count multiples of 5, of 11, and of 55 in [100,1500].
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Multiples of 5 in [100,1500]: $\lfloor1500/5\rfloor - \lfloor99/5\rfloor = 300 - 19 = 281$. Multiples of 11: $\lfloor1500/11\rfloor - \lfloor99/11\rfloor = 136 - 9 = 127$. Multiples of 55: $\lfloor1500/55\rfloor - \lfloor99/55\rfloor = 27 - 1 = 26$. By inclusion–exclusion: $281+127-26=382$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: 3