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08 Jan 2025, 02:33
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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. \(\frac{9}{37}\)
B. \(\frac{12}{37}\)
C. \(\frac{15}{37}\)
D. \(\frac{18}{37}\)
E. \(\frac{19}{37}\)
A. \(\frac{9}{37}\)
B. \(\frac{12}{37}\)
C. \(\frac{15}{37}\)
D. \(\frac{18}{37}\)
E. \(\frac{19}{37}\)
08 Jan 2025, 02:33
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Official Solution:
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. \(\frac{9}{37}\)
B. \(\frac{12}{37}\)
C. \(\frac{15}{37}\)
D. \(\frac{18}{37}\)
E. \(\frac{19}{37}\)
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 \(\frac{\text{odd numbers}}{\text{total}} = \frac{18}{37}\).
Answer: D
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. \(\frac{9}{37}\)
B. \(\frac{12}{37}\)
C. \(\frac{15}{37}\)
D. \(\frac{18}{37}\)
E. \(\frac{19}{37}\)
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 \(\frac{\text{odd numbers}}{\text{total}} = \frac{18}{37}\).
Answer: D
liberoasperiores
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22 Jan 2026, 03:31
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Is there a general rule for 3, 4 in consecutive integers?
Bunuel
