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How many even integers n, where 100 ≤ n ≤ 200, are divisible neither by seven nor by nine?
- a)m and n
- b)2m and 3n
- c)m and 2n
- d)2m and n
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
How many even integers n, where 100≤ n≤ 200,are divisible neithe...
Understanding the Problem
To find how many even integers n between 100 and 200 are divisible neither by 7 nor by 9, follow these steps:
Step 1: Identify the Range
- The even integers between 100 and 200 are:
100, 102, 104, ..., 200.
- This forms an arithmetic sequence with:
- First term (a) = 100
- Last term (l) = 200
- Common difference (d) = 2
Step 2: Count the Even Integers
- To find the count of even integers:
- n = (l - a) / d + 1
- n = (200 - 100) / 2 + 1 = 51
Step 3: Count the Even Integers Divisible by 7
- Even multiples of 7 between 100 and 200:
- First multiple = 104 (7 * 15)
- Last multiple = 196 (7 * 28)
- Count = (28 - 15) / 1 + 1 = 14
Step 4: Count the Even Integers Divisible by 9
- Even multiples of 9 between 100 and 200:
- First multiple = 108 (9 * 12)
- Last multiple = 198 (9 * 22)
- Count = (22 - 12) / 1 + 1 = 12
Step 5: Count the Overlap (Divisible by Both 7 and 9)
- Least Common Multiple (LCM) of 7 and 9 is 63.
- First even multiple of 63 = 126
- Last even multiple = 198
- Count = (198 - 126) / 126 + 1 = 2
Step 6: Apply Inclusion-Exclusion Principle
- Total divisible by either 7 or 9 = 14 + 12 - 2 = 24
Final Calculation
- Total even integers = 51
- Not divisible by either = 51 - 24 = 27
Thus, there are 27 even integers n, where 100 ≤ n ≤ 200, that are divisible neither by 7 nor by 9. The correct answer is option 'C': m and 2n.
To find how many even integers n between 100 and 200 are divisible neither by 7 nor by 9, follow these steps:
Step 1: Identify the Range
- The even integers between 100 and 200 are:
100, 102, 104, ..., 200.
- This forms an arithmetic sequence with:
- First term (a) = 100
- Last term (l) = 200
- Common difference (d) = 2
Step 2: Count the Even Integers
- To find the count of even integers:
- n = (l - a) / d + 1
- n = (200 - 100) / 2 + 1 = 51
Step 3: Count the Even Integers Divisible by 7
- Even multiples of 7 between 100 and 200:
- First multiple = 104 (7 * 15)
- Last multiple = 196 (7 * 28)
- Count = (28 - 15) / 1 + 1 = 14
Step 4: Count the Even Integers Divisible by 9
- Even multiples of 9 between 100 and 200:
- First multiple = 108 (9 * 12)
- Last multiple = 198 (9 * 22)
- Count = (22 - 12) / 1 + 1 = 12
Step 5: Count the Overlap (Divisible by Both 7 and 9)
- Least Common Multiple (LCM) of 7 and 9 is 63.
- First even multiple of 63 = 126
- Last even multiple = 198
- Count = (198 - 126) / 126 + 1 = 2
Step 6: Apply Inclusion-Exclusion Principle
- Total divisible by either 7 or 9 = 14 + 12 - 2 = 24
Final Calculation
- Total even integers = 51
- Not divisible by either = 51 - 24 = 27
Thus, there are 27 even integers n, where 100 ≤ n ≤ 200, that are divisible neither by 7 nor by 9. The correct answer is option 'C': m and 2n.
Community Answer
How many even integers n, where 100≤ n≤ 200,are divisible neithe...
Between 100 and 200 both included there are 51 even nos.
There are 7 even nos which are divisible by 7 and 6 nos which are divisible by 9 and 1 no divisible by both.
hence in total 51 - (7+6-1) = 39
There are 7 even nos which are divisible by 7 and 6 nos which are divisible by 9 and 1 no divisible by both.
hence in total 51 - (7+6-1) = 39
There is one more method through which we can find the answer.
Since we have to find even numbers, consider the numbers which are divisible by 14, 18 and 126 between 100 and 200.
These are 7, 6 and 1 respectively.
Since we have to find even numbers, consider the numbers which are divisible by 14, 18 and 126 between 100 and 200.
These are 7, 6 and 1 respectively.
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How many even integers n, where 100≤ n≤ 200,are divisible neither by seven nor by nine?a)m and nb)2m and 3nc)m and 2nd)2m and nCorrect answer is option 'C'. Can you explain this answer?
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