How many integers between 100 and 150, both inclusive, can be evenly divided by neither 3 nor 5?

### Concept & Logic To find the count of numbers divisible by "neither A nor B", we use Set Theory and the Principle of Inclusion-Exclusion. First, determine the total numbers in the range. Then find the count of numbers divisible by $A$, divisible by $B$, and divisible by both ($LCM$ of $A$ and $B$). The total numbers divisible by $A$ OR $B$ is: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ Finally, subtract this union from the total pool to get the "neither" count. ### Step-by-Step Solution * **Given:** Range is integers between $100$ and $150$, inclusive. Divisors are $3$ and $5$. * **Step 1: Find total integers in the range.** Total = $150 - 100 + 1 = 51$ numbers. * **Step 2: Calculate multiples of 3 ($n(3)$).** First multiple $\ge 100$ is $102$. Last $\le 150$ is $150$. Count = $\frac{150 - 102}{3} + 1 = \frac{48}{3} + 1 = 16 + 1 = 17$. * **Step 3: Calculate multiples of 5 ($n(5)$).** First multiple $\ge 100$ is $100$. Last $\le 150$ is $150$. Count = $\frac{150 - 100}{5} + 1 = \frac{50}{5} + 1 = 10 + 1 = 11$. * **Step 4: Calculate multiples of both 3 and 5 ($n(15)$).** First multiple of $15 \ge 100$ is $105$. Last $\le 150$ is $150$. Count = $\frac{150 - 105}{15} + 1 = \frac{45}{15} + 1 = 3 + 1 = 4$. * **Step 5: Apply Inclusion-Exclusion.** Numbers divisible by $3$ or $5$: $n(3 \cup 5) = 17 + 11 - 4 = 24$. * **Step 6: Find the "neither" count.** Neither = Total - $n(3 \cup 5) = 51 - 24 = 27$. ### Exam Strategy & Shortcut Use the quotient method for fast range counts: Let $F(N, X) = \lfloor \frac{N}{X} \rfloor$ Multiples in range $100$ to $150$ for divisor $X$ = $F(150, X) - F(99, X)$. * $n(3) = 50 - 33 = 17$ * $n(5) = 30 - 19 = 11$ * $n(15) = 10 - 6 = 4$ Divisible by $3$ or $5 = 17 + 11 - 4 = 24$. Total numbers = $150 - 99 = 51$. Neither = $51 - 24 = 27$. This relies entirely on fast division rather than setting up A.P. formulas. ### Common Pitfall The word "inclusive" means both boundaries are part of the count. Many students simply subtract $150 - 100 = 50$ total numbers, omitting the $+1$. This throws off the final arithmetic and leads to choosing option $26$ instead of $27$. ### Final Answer **Therefore, the correct answer is 27.**