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D
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Difficulty:
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Question Stats:
56% (02:26) correct
44%
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If k is an integer from 48 to 84, inclusive, what is the probability that (k - 1)(k + 1)(k + 3) is divisible by 48?
A. 9/37
B. 12/37
C. 15/37
D. 18/37
E. 19/37
If k is an integer from 48 to 84, inclusive, what is the probability that (k - 1)(k + 1)(k + 3) is divisible by 48?
A. 9/37
B. 12/37
C. 15/37
D. 18/37
E. 19/37
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20 Dec 2024, 10:00
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Bunuel
GMAT Club's Official Explanation:
Consider two cases:
• When k is even, (k - 1)(k + 1)(k + 3) will be odd * odd * odd, which is odd, so it will not be divisible by an even number like 48.
• When k is odd, (k - 1)(k + 1)(k + 3) represents the product of three consecutive even integers, one of which will be divisible by 4, making the product divisible by 2 * 2 * 4 = 16. Furthermore, one of the three consecutive even integers must be divisible by 3. Thus, (k - 1)(k + 1)(k + 3) will be divisible by 16 * 3 = 48 for any odd value of k.
Since there are a total of 37 integers from 48 to 84, inclusive, out of which 18 are odd, the probability that (k - 1)(k + 1)(k + 3) is divisible by 48 is (odd numbers)/(total) = 18/37.
Answer: D.
General Discussion
If k is an integer from 48 to 84, inclusive, what is the probability that (k - 1)(k + 1)(k + 3) is divisible by 48?
(k - 1)(k + 1)(k + 3) is divisible by 3 since one of (k-1), (k+1) or (k+3) is divisible by 3.
Case 1: k is odd
k-1, k+1, k+3 are all even at least one of them will be divisible by 4.
(k-1)(k+1)(k+3) is divisible by 3*2*2*4 = 48
Case 2: k is even
k-1, k+1, k+3 are all odd
(k-1)(k+1)(k+3) is NOT divisible by 48
Total numbers from 48 to 84, inclusive = 84-48+1 = 37
Odd numbers = {49,51,.....83}: Number of odd numbers = (83-49)/2 + 1 = 18
The probability that (k - 1)(k + 1)(k + 3) is divisible by 48 = 18/37
IMO D
(k - 1)(k + 1)(k + 3) is divisible by 3 since one of (k-1), (k+1) or (k+3) is divisible by 3.
Case 1: k is odd
k-1, k+1, k+3 are all even at least one of them will be divisible by 4.
(k-1)(k+1)(k+3) is divisible by 3*2*2*4 = 48
Case 2: k is even
k-1, k+1, k+3 are all odd
(k-1)(k+1)(k+3) is NOT divisible by 48
Total numbers from 48 to 84, inclusive = 84-48+1 = 37
Odd numbers = {49,51,.....83}: Number of odd numbers = (83-49)/2 + 1 = 18
The probability that (k - 1)(k + 1)(k + 3) is divisible by 48 = 18/37
IMO D
