Question:
The number of positive integers n in the range \( 12 \leq n \leq 40 \) such that the product (n -1)*(n - 2)*…*3*2*1 is not divisible by n is
The number of positive integers n in the range \( 12 \leq n \leq 40 \) such that the product (n -1)*(n - 2)*…*3*2*1 is not divisible by n is
Show Hint
If n can be written as a product of two different smaller numbers, both of them sit inside the factorial. Which numbers cannot be written like that?
Updated On: Nov 7, 2026
- 5
- 7
- 13
- 14
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Understand the question:
We look at n from 12 to 40. For each n we form \((n-1)! = (n-1)(n-2) \cdots 3 \cdot 2 \cdot 1\) and ask whether n divides it. We must count the n for which n does NOT divide \((n-1)!\).
Step 2: Case 1: n is prime:
If n is prime, none of the numbers 1, 2, ..., n-1 is a multiple of n, and a prime cannot be split into smaller factors. So n does not divide \((n-1)!\).
Every prime in the range is therefore a valid n.
Step 3: Case 2: n is composite, with two different factors:
Suppose \(n = a \times b\) with \(1 < a < b < n\). Both a and b appear among the factors 1, 2, ..., n-1, and they are different numbers.
So the product \(a \times b = n\) is already inside \((n-1)!\), and n divides \((n-1)!\). These n are not counted.
Step 4: Case 3: n is the square of a prime:
If \(n = p^2\), the factors a and b above would be equal, so we check separately.
In our range the only such n is 25 (since 9 and 49 are outside). Here \(24!\) contains both 5 and 10, and \(5 \times 10 = 50\) has \(5^2\) as a factor. So 25 divides \(24!\) and is not counted.
Step 5: Count the primes between 12 and 40:
The primes are 13, 17, 19, 23, 29, 31, 37.
(Check: 11 is below 12, and 41 is above 40. The numbers 15, 21, 27, 33, 35, 39 are odd but have factors 3 or 5 or 7, so they are not prime.)
That is 7 primes, so there are 7 values of n.
Step 6: Why the other options are wrong:
Option 1 (5) undercounts, because the range holds 7 primes.
Option 3 (13) and option 4 (14) are far too big: of the 29 numbers from 12 to 40, only 7 are prime and all the others divide \((n-1)!\).
Final Answer:
The numbers that do not divide \((n-1)!\) are exactly the primes in the range, and there are 7 of them. This is option 2. \[ \boxed{7 \; \text{(option 2)}} \]
We look at n from 12 to 40. For each n we form \((n-1)! = (n-1)(n-2) \cdots 3 \cdot 2 \cdot 1\) and ask whether n divides it. We must count the n for which n does NOT divide \((n-1)!\).
Step 2: Case 1: n is prime:
If n is prime, none of the numbers 1, 2, ..., n-1 is a multiple of n, and a prime cannot be split into smaller factors. So n does not divide \((n-1)!\).
Every prime in the range is therefore a valid n.
Step 3: Case 2: n is composite, with two different factors:
Suppose \(n = a \times b\) with \(1 < a < b < n\). Both a and b appear among the factors 1, 2, ..., n-1, and they are different numbers.
So the product \(a \times b = n\) is already inside \((n-1)!\), and n divides \((n-1)!\). These n are not counted.
Step 4: Case 3: n is the square of a prime:
If \(n = p^2\), the factors a and b above would be equal, so we check separately.
In our range the only such n is 25 (since 9 and 49 are outside). Here \(24!\) contains both 5 and 10, and \(5 \times 10 = 50\) has \(5^2\) as a factor. So 25 divides \(24!\) and is not counted.
Step 5: Count the primes between 12 and 40:
The primes are 13, 17, 19, 23, 29, 31, 37.
(Check: 11 is below 12, and 41 is above 40. The numbers 15, 21, 27, 33, 35, 39 are odd but have factors 3 or 5 or 7, so they are not prime.)
That is 7 primes, so there are 7 values of n.
Step 6: Why the other options are wrong:
Option 1 (5) undercounts, because the range holds 7 primes.
Option 3 (13) and option 4 (14) are far too big: of the 29 numbers from 12 to 40, only 7 are prime and all the others divide \((n-1)!\).
Final Answer:
The numbers that do not divide \((n-1)!\) are exactly the primes in the range, and there are 7 of them. This is option 2. \[ \boxed{7 \; \text{(option 2)}} \]
Was this answer helpful?
0
0
Top CAT Quantitative Aptitude Questions
- In a triangle ABC,AB=AC=8cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area,in sq.cm,of the overlapping region between the two circles is
- The simple interest accrued on an amount of Rs.25,000 at the end of three years is Rs.7,500. What would be the compound interest accrued on the same amount at the same rate in the same period?
- What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?
- A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?
- The ratio of the ages of a husband and his wife when they got married was 6:5. 4 years and 6 years after their marriage they had their 1st and 2nd children. The sum of the present ages of the husband and wife is 6.4 times the sum of the present ages of their children. The average age of the family at present is 18.5 years. Find the ratio of the ages of the husband and wife when their second child was born.
View More Questions
Top CAT Number System Questions
- Consider a sequence \( S \) whose \( n \)th term \( T_n \) is defined as \( T_n = 1 + \frac{3}{n} \), where \( n = 1, 2, \ldots \). Find the product of all the consecutive terms of \( S \) starting from the 4th term to the 60th term.
- Let \( P = \{2, 3, 4, \ldots, 100\} \) and \( Q = \{101, 102, 103, \ldots, 200\} \). How many elements of \( Q \) are there such that they do not have any element of \( P \) as a factor?
- Find the remainder when \( 2^{1040} \) is divided by 131.
- \( p \) is a prime and \( m \) is a positive integer. How many solutions exist for the equation \[ p^5 - p = (m^2 + m + 6)(p - 1)? \]
- A certain number written in a certain base is 144. Which of the following is always true?
View More Questions
Top CAT Questions
- Compare and contrast between Portia and Jessica characters in Merchant of Venice.
- In a triangle ABC,AB=AC=8cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area,in sq.cm,of the overlapping region between the two circles is
- The simple interest accrued on an amount of Rs.25,000 at the end of three years is Rs.7,500. What would be the compound interest accrued on the same amount at the same rate in the same period?
- What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?
- A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?
View More Questions