How many integers from 700 to 950 inclusive are neither divisible by 3 nor divisible by 7?

Introduction / Context:
This counting problem uses the inclusion exclusion principle to find how many numbers in a given range are not divisible by certain divisors. The range is from 700 to 950 inclusive, and we must exclude numbers divisible by either 3 or 7. Such problems are common in aptitude exams and reinforce understanding of multiples and overlaps between sets.

Given Data / Assumptions:
Range of integers: 700 to 950 inclusive.We want numbers not divisible by 3 and not divisible by 7.We will count and subtract those divisible by 3 or 7 using inclusion exclusion.

Concept / Approach:
First, count the total numbers in the range. Then count how many are divisible by 3 and how many are divisible by 7. These counts overlap for numbers divisible by both 3 and 7, that is, by 21. The count of numbers divisible by 3 or 7 is count(3) + count(7) − count(21). Finally, subtract this from the total to find numbers divisible by neither.

Step-by-Step Solution:
Step 1: Total numbers from 700 to 950 inclusive: 950 − 700 + 1 = 251.Step 2: Count multiples of 3 in this range.Step 3: First multiple of 3 not less than 700 is 702 (3 × 234).Step 4: Last multiple of 3 not greater than 950 is 948 (3 × 316).Step 5: Number of multiples of 3 is 316 − 234 + 1 = 83.Step 6: Count multiples of 7 in the range.Step 7: First multiple of 7 not less than 700 is 700 (7 × 100).Step 8: Last multiple of 7 not greater than 950 is 945 (7 × 135).Step 9: Number of multiples of 7 is 135 − 100 + 1 = 36.Step 10: Count multiples of 21 (LCM of 3 and 7) to correct for double counting.Step 11: First multiple of 21 not less than 700 is 714 (21 × 34).Step 12: Last multiple of 21 not greater than 950 is 945 (21 × 45).Step 13: Number of multiples of 21 is 45 − 34 + 1 = 12.Step 14: Numbers divisible by 3 or 7: 83 + 36 − 12 = 107.Step 15: Numbers divisible by neither 3 nor 7: 251 − 107 = 144.

Verification / Alternative check:
As a quick sanity check, note that about one third of numbers are multiples of 3 and about one seventh are multiples of 7, with some overlap. From 251 numbers, having roughly 107 excluded is reasonable, leaving about 144, which matches the exact computed value. Spot checking a few numbers near boundaries confirms the first and last multiples for each divisor are chosen correctly.

Why Other Options Are Wrong:
The option 107 is the count of numbers that are divisible by 3 or 7, not the count of numbers that avoid both. The values 141, 145 and other close numbers would reflect minor miscalculations in counting multiples or applying inclusion exclusion incorrectly, such as forgetting to subtract the multiples of 21.

Common Pitfalls:
Common mistakes include forgetting to add 1 when counting terms inclusively, misidentifying the first or last multiples in the range, or neglecting to subtract the overlap of multiples of 21. Systematically computing each count and writing out each step reduces these errors.

Final Answer:
The number of integers from 700 to 950 that are neither divisible by 3 nor by 7 is 144.