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How many even integers n, where 100 ≤ n ≤ 200, are divisible neither by seven nor by nine?
- a)40
- b)37
- c)39
- d)38
Correct answer is option 'C'. Can you explain this answer?
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How many even integers n, where 100≤ n≤ 200,are divisible neithe...
- We need to find even integers n between 100 and 200.
- The even integers in this range are: 100, 102, ..., 200. There are 51 even numbers.
- We check for divisibility:
- Even numbers divisible by 7: 104, 112, 120, 126, 140, 154, 168.
- Even numbers divisible by 9: 108, 126, 144, 162, 180, 198.
- Even numbers divisible by both (14): 126.
- Using inclusion-exclusion:
- Total divisible = 7 + 6 - 1 = 12.
- Thus, even integers not divisible by either = 51 - 12 = 39.
- The even integers in this range are: 100, 102, ..., 200. There are 51 even numbers.
- We check for divisibility:
- Even numbers divisible by 7: 104, 112, 120, 126, 140, 154, 168.
- Even numbers divisible by 9: 108, 126, 144, 162, 180, 198.
- Even numbers divisible by both (14): 126.
- Using inclusion-exclusion:
- Total divisible = 7 + 6 - 1 = 12.
- Thus, even integers not divisible by either = 51 - 12 = 39.
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How many even integers n, where 100≤ n≤ 200,are divisible neithe...
Problem Overview
We need to find the count of even integers n within the range 100 ≤ n ≤ 200 that are neither divisible by 7 nor by 9.
Step 1: Identify the Even Integers
- The even integers between 100 and 200 are: 100, 102, 104, ..., 200.
- This forms an arithmetic sequence where:
- First term (a) = 100
- Common difference (d) = 2
- Last term (l) = 200
- To find the total number of even integers (n):
- Formula: n = (l - a)/d + 1
- Calculation: n = (200 - 100)/2 + 1 = 51
Step 2: Count Divisible by 7
- The even integers divisible by 7 in this range are: 105, 112, 119, ..., 196.
- The first even multiple of 7 is 105 and the last is 196.
- To find total:
- Sequence: 112, 126, 140, 154, 168, 182, 196
- Number of terms = (196 - 112)/14 + 1 = 7
Step 3: Count Divisible by 9
- The even integers divisible by 9 are: 108, 126, 144, ..., 198.
- The first even multiple of 9 is 108 and the last is 198.
- To find total:
- Sequence: 108, 126, 144, 162, 180, 198
- Number of terms = (198 - 108)/18 + 1 = 6
Step 4: Count Divisible by Both 7 and 9
- The least common multiple (LCM) of 7 and 9 is 63.
- The even multiples of 63 in this range are: 126.
- Count = 1
Step 5: Apply Inclusion-Exclusion Principle
- Total divisible by 7 or 9 = (count of 7) + (count of 9) - (count of both)
- Total = 7 + 6 - 1 = 12
Final Count
- Even integers not divisible by 7 or 9 = Total even integers - Total divisible by 7 or 9
- Result = 51 - 12 = 39
Thus, the answer is option 'C' which is 39.
We need to find the count of even integers n within the range 100 ≤ n ≤ 200 that are neither divisible by 7 nor by 9.
Step 1: Identify the Even Integers
- The even integers between 100 and 200 are: 100, 102, 104, ..., 200.
- This forms an arithmetic sequence where:
- First term (a) = 100
- Common difference (d) = 2
- Last term (l) = 200
- To find the total number of even integers (n):
- Formula: n = (l - a)/d + 1
- Calculation: n = (200 - 100)/2 + 1 = 51
Step 2: Count Divisible by 7
- The even integers divisible by 7 in this range are: 105, 112, 119, ..., 196.
- The first even multiple of 7 is 105 and the last is 196.
- To find total:
- Sequence: 112, 126, 140, 154, 168, 182, 196
- Number of terms = (196 - 112)/14 + 1 = 7
Step 3: Count Divisible by 9
- The even integers divisible by 9 are: 108, 126, 144, ..., 198.
- The first even multiple of 9 is 108 and the last is 198.
- To find total:
- Sequence: 108, 126, 144, 162, 180, 198
- Number of terms = (198 - 108)/18 + 1 = 6
Step 4: Count Divisible by Both 7 and 9
- The least common multiple (LCM) of 7 and 9 is 63.
- The even multiples of 63 in this range are: 126.
- Count = 1
Step 5: Apply Inclusion-Exclusion Principle
- Total divisible by 7 or 9 = (count of 7) + (count of 9) - (count of both)
- Total = 7 + 6 - 1 = 12
Final Count
- Even integers not divisible by 7 or 9 = Total even integers - Total divisible by 7 or 9
- Result = 51 - 12 = 39
Thus, the answer is option 'C' which is 39.
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How many even integers n, where 100≤ n≤ 200,are divisible neither by seven nor by nine?a)40b)37c)39d)38Correct answer is option 'C'. Can you explain this answer?
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