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A number N is given by = 25 x 38 x 42 x 53 x 6 x 72
If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where 'n' is a natural number less than 20. Find the value of m+n.
If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where 'n' is a natural number less than 20. Find the value of m+n.
Correct answer is '12'. Can you explain this answer?
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A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor...
Let us start by writing the N in empirical form which is 25 x 38 x 24 x 53 x 2 x 3 x 72
Which equals 210 x 39 x 53 x 72
Let A = number of factors not divisible by 9
B = Number of Factors which are odd
We have to find
(A∩B) = Number of factors not divisible by 9 and are odd. It implies it does not have 2 as a factor and the highest power of 3 can be 1. Such type of factors are = (1+1)(3+1)(2+1) = 24
A = Number of factors that are not divisible by 9. Thus the maximum power of 3 they can have is 1. Such number are = (10+1)(1+1)(3+1)(2+1) = 264
Probability = 24/264 = 1/11
m+n = 12.
Which equals 210 x 39 x 53 x 72
Let A = number of factors not divisible by 9
B = Number of Factors which are odd
We have to find
(A∩B) = Number of factors not divisible by 9 and are odd. It implies it does not have 2 as a factor and the highest power of 3 can be 1. Such type of factors are = (1+1)(3+1)(2+1) = 24
A = Number of factors that are not divisible by 9. Thus the maximum power of 3 they can have is 1. Such number are = (10+1)(1+1)(3+1)(2+1) = 264
Probability = 24/264 = 1/11
m+n = 12.
Most Upvoted Answer
A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor...
To find the probability that a factor of N is an odd number and not divisible by 9, we need to first determine the total number of factors of N.
Finding Factors of N:
N = 25 x 38 x 42 x 53 x 6 x 72
To find the factors of N, we can break it down into its prime factors:
N = (5^2) x (2 x 19) x (2 x 3^1 x 7) x (53) x (2 x 3) x (2^3 x 3^2)
N = 2^7 x 3^4 x 5^2 x 7 x 19 x 53
To find the total number of factors, we can use the formula:
Total number of factors = (a+1)(b+1)(c+1)... where a, b, c are the exponents of the prime factors.
In this case, the exponents are:
a = 7 (for 2)
b = 4 (for 3)
c = 2 (for 5)
d = 1 (for 7)
e = 1 (for 19)
f = 1 (for 53)
Total number of factors = (7+1)(4+1)(2+1)(1+1)(1+1)(1+1) = 8 x 5 x 3 x 2 x 2 x 2 = 480
Finding Odd Factors not Divisible by 9:
Next, we need to find the number of odd factors of N that are not divisible by 9.
To be an odd factor, the factors must include at least one odd prime factor (3, 5, 7, 19, or 53).
Since the factor cannot be divisible by 9, it cannot include the prime factor 3 raised to a power greater than 1.
The possible combinations of odd prime factors that are not divisible by 9 are:
- (5) or (5^2) or (5^2 x 7) or (5^2 x 7 x 19) or (5^2 x 7 x 19 x 53)
- (7) or (7 x 19) or (7 x 19 x 53)
Thus, the total number of odd factors not divisible by 9 is 5 + 3 = 8.
Finding the Probability:
The probability that a factor of N is an odd number and not divisible by 9 can be calculated as:
Probability = Number of favorable outcomes / Total number of outcomes
Number of favorable outcomes = 8
Total number of outcomes = 480
Probability = 8/480 = 1/60
Therefore, the value of m*n is 1*60 = 60, not 12.
Finding Factors of N:
N = 25 x 38 x 42 x 53 x 6 x 72
To find the factors of N, we can break it down into its prime factors:
N = (5^2) x (2 x 19) x (2 x 3^1 x 7) x (53) x (2 x 3) x (2^3 x 3^2)
N = 2^7 x 3^4 x 5^2 x 7 x 19 x 53
To find the total number of factors, we can use the formula:
Total number of factors = (a+1)(b+1)(c+1)... where a, b, c are the exponents of the prime factors.
In this case, the exponents are:
a = 7 (for 2)
b = 4 (for 3)
c = 2 (for 5)
d = 1 (for 7)
e = 1 (for 19)
f = 1 (for 53)
Total number of factors = (7+1)(4+1)(2+1)(1+1)(1+1)(1+1) = 8 x 5 x 3 x 2 x 2 x 2 = 480
Finding Odd Factors not Divisible by 9:
Next, we need to find the number of odd factors of N that are not divisible by 9.
To be an odd factor, the factors must include at least one odd prime factor (3, 5, 7, 19, or 53).
Since the factor cannot be divisible by 9, it cannot include the prime factor 3 raised to a power greater than 1.
The possible combinations of odd prime factors that are not divisible by 9 are:
- (5) or (5^2) or (5^2 x 7) or (5^2 x 7 x 19) or (5^2 x 7 x 19 x 53)
- (7) or (7 x 19) or (7 x 19 x 53)
Thus, the total number of odd factors not divisible by 9 is 5 + 3 = 8.
Finding the Probability:
The probability that a factor of N is an odd number and not divisible by 9 can be calculated as:
Probability = Number of favorable outcomes / Total number of outcomes
Number of favorable outcomes = 8
Total number of outcomes = 480
Probability = 8/480 = 1/60
Therefore, the value of m*n is 1*60 = 60, not 12.
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A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. Can you explain this answer?
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A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. 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 A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. Can you explain this answer?.
A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. 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 A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A number N is given by = 25 x 38 x 42 x 53 x 6 x 72If a certain factor of N is not divisible by 9, the probability that the factor is an odd number is m/n where n is a natural number less than 20. Find the value of m+n.Correct answer is '12'. Can you explain this answer?.
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