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09 Nov 2023, 00:39
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How many distinct positive integers \(d\) exist such that, when 48 is divided by \(d\), the remainder is \(d - 4\)?
A. Two
B. Three
C. Four
D. Five
E. Six
A. Two
B. Three
C. Four
D. Five
E. Six
09 Nov 2023, 00:39
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Official Solution:
How many distinct positive integers \(d\) exist such that, when 48 is divided by \(d\), the remainder is \(d - 4\)?
A. Two
B. Three
C. Four
D. Five
E. Six
Dividing 48 by \(d\) and obtaining a remainder of \(d - 4\) can be expressed as \(48 = qd + (d - 4)\). Note that since the remainder must be non-negative, we have \(d - 4 \geq 0\), which implies \(d \geq 4\).
Rewriting \(48 = qd + (d - 4)\) leads to \(q + 1= \frac{52}{d}\). Here, since \(q + 1\) is an integer, \(\frac{52}{d}\) must also be an integer, meaning \(d\) is a factor of 52. The factors of 52 are 1, 2, 4, 13, 26, and 52. Given that \(d \geq 4\), the valid values for \(d\) are 4, 13, 26, and 52.
Therefore, there are four possible values for \(d\).
Answer: C
How many distinct positive integers \(d\) exist such that, when 48 is divided by \(d\), the remainder is \(d - 4\)?
A. Two
B. Three
C. Four
D. Five
E. Six
Dividing 48 by \(d\) and obtaining a remainder of \(d - 4\) can be expressed as \(48 = qd + (d - 4)\). Note that since the remainder must be non-negative, we have \(d - 4 \geq 0\), which implies \(d \geq 4\).
Rewriting \(48 = qd + (d - 4)\) leads to \(q + 1= \frac{52}{d}\). Here, since \(q + 1\) is an integer, \(\frac{52}{d}\) must also be an integer, meaning \(d\) is a factor of 52. The factors of 52 are 1, 2, 4, 13, 26, and 52. Given that \(d \geq 4\), the valid values for \(d\) are 4, 13, 26, and 52.
Therefore, there are four possible values for \(d\).
Answer: C
Hey I followed the method you have shared however, if we have taken d to be 13, then the remainder when 48 is divided by 13 is 5 but d-4 = 13-4 =9. The rest holds for 4, 26, and 52. Am I missing something here?
