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21 May 2022, 10:32
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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
M40-07
A. Two
B. Three
C. Four
D. Five
E. Six
M40-07
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09 Nov 2023, 00:40
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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
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
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
General Discussion
21 May 2022, 12:27
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48 divided by d yields a remainder of d - 4 can be written mathematically as: \(\frac{48}{d} = Q\frac{d-4}{d}\) where Q = Quotient.
In other words \((Q*d)+d-4 = 48\)
Which simplified is: \(Qd + d = 52\)
Now the question simply becomes: what are the factors of 52 which are 4 or greater (as the lowest value that d-4 can be is 0 given that it is a remainder).
Factors of 52:
\(1*52\)
\(2*26\)
\(4*13\)
Factors of 52 4 or greater: \(4, 13, 26\) and \(52\), all of which when dividing 48 leave a remainder of \(d-4\).
Answer is therefore C, four.
To double check:
4: \(\frac{48}{4}= 12\) with a remainder of 0. \(4-4=0\)
13: \(\frac{48}{13}= 3\) with a remainder of 9. \(13-4=9\)
26: \(\frac{48}{26}= 1\) with a remainder of 22. \(26-4=22\)
\(52: \) \(\frac{48}{52} = 0\) with a remainder of 48. \(52-4=48\)
Answer C
In other words \((Q*d)+d-4 = 48\)
Which simplified is: \(Qd + d = 52\)
Now the question simply becomes: what are the factors of 52 which are 4 or greater (as the lowest value that d-4 can be is 0 given that it is a remainder).
Factors of 52:
\(1*52\)
\(2*26\)
\(4*13\)
Factors of 52 4 or greater: \(4, 13, 26\) and \(52\), all of which when dividing 48 leave a remainder of \(d-4\).
Answer is therefore C, four.
To double check:
4: \(\frac{48}{4}= 12\) with a remainder of 0. \(4-4=0\)
13: \(\frac{48}{13}= 3\) with a remainder of 9. \(13-4=9\)
26: \(\frac{48}{26}= 1\) with a remainder of 22. \(26-4=22\)
\(52: \) \(\frac{48}{52} = 0\) with a remainder of 48. \(52-4=48\)
Answer C
