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If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?
1) 2 is not a factor of n. ?
2) 3 is not a factor of n.
- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If n is a positive integer and r is the remainder when (n – 1)(n...
(1) n is not divisible by 2. Insufficient on its own, but this statement says that n=odd--> n−1 and n+1 are consecutive even integers --> (n−1)(n+1) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...).
(2) n is not divisible by 3. Insufficient on its own, but form this statement either n−1 or n+1 must be divisible by 3 (as n−1, n, and n+1 are consecutive integers, so one of them must be divisible by 3, we are told that it's not n, hence either n−1 or n+1).
(1)+(2) From (1) (n−1)(n+1) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by 8∗3=24, which means that remainder upon division (n−1)(n+1) by 24 will be 0. Sufficient.
(2) n is not divisible by 3. Insufficient on its own, but form this statement either n−1 or n+1 must be divisible by 3 (as n−1, n, and n+1 are consecutive integers, so one of them must be divisible by 3, we are told that it's not n, hence either n−1 or n+1).
(1)+(2) From (1) (n−1)(n+1) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by 8∗3=24, which means that remainder upon division (n−1)(n+1) by 24 will be 0. Sufficient.
Most Upvoted Answer
If n is a positive integer and r is the remainder when (n – 1)(n...
We can use the division algorithm to write:
n = dq + r
where d is a positive integer and 0 ≤ r < q.="" />
Rearranging this equation, we get:
r = n - dq
We know that n is a multiple of 13, so we can write:
n = 13k
where k is a positive integer.
Substituting this into the equation above, we get:
r = 13k - dq
Since r is less than q, we can write:
r = q - s
where s is a positive integer.
Substituting this into the equation above, we get:
q - s = 13k - dq
Simplifying, we get:
dq + s = 13k - q
Since d and q are positive integers, we know that dq + s is greater than q. Therefore, we can write:
dq + s = q + 13m
where m is a positive integer.
Substituting this into the equation above, we get:
q + 13m = 13k - q
Simplifying, we get:
2q + 13m = 13k
This shows that 2q is a multiple of 13. Since 13 is a prime number, this means that q must be a multiple of 13.
Therefore, the remainder r when n is divided by q must be less than q, and it must also be a multiple of 13. The only multiple of 13 that is less than q is 0. Therefore, the remainder r must be 0.
n = dq + r
where d is a positive integer and 0 ≤ r < q.="" />
Rearranging this equation, we get:
r = n - dq
We know that n is a multiple of 13, so we can write:
n = 13k
where k is a positive integer.
Substituting this into the equation above, we get:
r = 13k - dq
Since r is less than q, we can write:
r = q - s
where s is a positive integer.
Substituting this into the equation above, we get:
q - s = 13k - dq
Simplifying, we get:
dq + s = 13k - q
Since d and q are positive integers, we know that dq + s is greater than q. Therefore, we can write:
dq + s = q + 13m
where m is a positive integer.
Substituting this into the equation above, we get:
q + 13m = 13k - q
Simplifying, we get:
2q + 13m = 13k
This shows that 2q is a multiple of 13. Since 13 is a prime number, this means that q must be a multiple of 13.
Therefore, the remainder r when n is divided by q must be less than q, and it must also be a multiple of 13. The only multiple of 13 that is less than q is 0. Therefore, the remainder r must be 0.
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Community Answer
If n is a positive integer and r is the remainder when (n – 1)(n...
Correct answer will be option C
how?
statement-1 put any odd value of N in the equation, you will get different remainder.out
statement-2 put any value of N other than a number multiple of 3 in the equation.remainder will be different.
combining both we can say N is not divisible by either 2 or 3 that means any value of N=6K is not possible.
But N can be 6K+1 or 6K-1
so (N-1)(N+1)=6k*(6k+2)=12k(3k+1)
now see if k is odd 3k+1 will be even ,that implies
12*odd*even= 12*even ,that implies that even number is a multiple of 2 ,so 12*2 =24 will be the minimum value of the equation, which when divided by 24 will leave 0 remainder
and same you can prove for N=6K-1.
so answer is C
how?
statement-1 put any odd value of N in the equation, you will get different remainder.out
statement-2 put any value of N other than a number multiple of 3 in the equation.remainder will be different.
combining both we can say N is not divisible by either 2 or 3 that means any value of N=6K is not possible.
But N can be 6K+1 or 6K-1
so (N-1)(N+1)=6k*(6k+2)=12k(3k+1)
now see if k is odd 3k+1 will be even ,that implies
12*odd*even= 12*even ,that implies that even number is a multiple of 2 ,so 12*2 =24 will be the minimum value of the equation, which when divided by 24 will leave 0 remainder
and same you can prove for N=6K-1.
so answer is C
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If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer?.
If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer?.
Solutions for If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n.a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT.
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