CAT Exam > CAT Questions > Let P be the product of the first 1...
Start Learning for Free
Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k.
- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Let P be the product of the first 100 positive odd integers....
Note that the product of the first 100 positive odd integers can be written as
Hence, we seek the number of threes in 200! decreased by the number of threes in 100!
There are
View all questions of this test
Hence, we seek the number of threes in 200! decreased by the number of threes in 100!
There are
Therefore, we have a total of 97-48=049 threes.
Most Upvoted Answer
Let P be the product of the first 100 positive odd integers....
To find the largest integer k such that P is divisible by 3k, we need to determine the highest power of 3 that divides the product of the first 100 positive odd integers.
Product of the first 100 positive odd integers:
P = 1 * 3 * 5 * 7 * ... * 195 * 197 * 199
We notice that every odd number can be written as 2n+1, where n is a non-negative integer. Substituting this expression for each odd number in the product, we have:
P = (2*0+1) * (2*1+1) * (2*2+1) * ... * (2*99+1)
Simplifying the expression, we get:
P = 1 * 3 * 5 * 7 * ... * 195 * 197 * 199
Since we are looking for the largest integer k such that P is divisible by 3k, we need to find the highest power of 3 that divides P.
To do this, we can count the number of factors of 3 in the product. We know that every third number in the product is divisible by 3. So, let's count the number of multiples of 3 in the product.
Counting the multiples of 3:
3, 6, 9, 12, ... , 195
We can observe that there are 66 multiples of 3 in the product.
Since we are counting the number of factors of 3, we need to divide this count by 2. This is because every third number is divisible by 3, but we are counting pairs of factors of 3. So, we have a total of 33 pairs of factors of 3 in the product.
Therefore, the highest power of 3 that divides P is 3^33.
To find the largest integer k such that P is divisible by 3k, we need to divide the highest power of 3 by 3.
33/3 = 11
Hence, the largest integer k is 11.
Therefore, the correct answer is option B) 49.
Product of the first 100 positive odd integers:
P = 1 * 3 * 5 * 7 * ... * 195 * 197 * 199
We notice that every odd number can be written as 2n+1, where n is a non-negative integer. Substituting this expression for each odd number in the product, we have:
P = (2*0+1) * (2*1+1) * (2*2+1) * ... * (2*99+1)
Simplifying the expression, we get:
P = 1 * 3 * 5 * 7 * ... * 195 * 197 * 199
Since we are looking for the largest integer k such that P is divisible by 3k, we need to find the highest power of 3 that divides P.
To do this, we can count the number of factors of 3 in the product. We know that every third number in the product is divisible by 3. So, let's count the number of multiples of 3 in the product.
Counting the multiples of 3:
3, 6, 9, 12, ... , 195
We can observe that there are 66 multiples of 3 in the product.
Since we are counting the number of factors of 3, we need to divide this count by 2. This is because every third number is divisible by 3, but we are counting pairs of factors of 3. So, we have a total of 33 pairs of factors of 3 in the product.
Therefore, the highest power of 3 that divides P is 3^33.
To find the largest integer k such that P is divisible by 3k, we need to divide the highest power of 3 by 3.
33/3 = 11
Hence, the largest integer k is 11.
Therefore, the correct answer is option B) 49.
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer?
Question Description
Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer?.
Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer?.
Solutions for Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer?, a detailed solution for Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3k. a) 24 b) 49 c) 78 d) 97 Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup