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07 Nov 2023, 07:09
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When a positive integer \(x\) is divided by \(d\), the remainder is 241, and when \(2x\) is divided by \(d\), the remainder is 112. What is the value of \(d\)?
A. 129
B. 258
C. 353
D. 370
E. 706
A. 129
B. 258
C. 353
D. 370
E. 706
07 Nov 2023, 07:09
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Official Solution:
When a positive integer \(x\) is divided by \(d\), the remainder is 241, and when \(2x\) is divided by \(d\), the remainder is 112. What is the value of \(d\)?
A. 129
B. 258
C. 353
D. 370
E. 706
\(x\) divided by \(d\), yielding a remainder of 241, can be expressed as \(x = qd + 241\). Observe that, since the divisor must always be greater than the remainder, then \(d > 241\).
Similarly, \(2x\) divided by \(d\), yielding a remainder of 112, can be expressed as \(2x = pd + 112\).
Multiplying the first equation by 2, we get: \(2x = 2qd + 482\).
Subtracting the second equation from the above gives: \(0 = 2qd + 482 - (pd + 112)\), which simplifies to \(p - 2q = \frac{370}{d}\). Since \(p - 2q\) is an integer, \(\frac{370}{d}\) must also be an integer, implying that \(d\) is a factor of 370. However, given that \(d > 241\), it can only be 370 itself.
Answer: D
When a positive integer \(x\) is divided by \(d\), the remainder is 241, and when \(2x\) is divided by \(d\), the remainder is 112. What is the value of \(d\)?
A. 129
B. 258
C. 353
D. 370
E. 706
\(x\) divided by \(d\), yielding a remainder of 241, can be expressed as \(x = qd + 241\). Observe that, since the divisor must always be greater than the remainder, then \(d > 241\).
Similarly, \(2x\) divided by \(d\), yielding a remainder of 112, can be expressed as \(2x = pd + 112\).
Multiplying the first equation by 2, we get: \(2x = 2qd + 482\).
Subtracting the second equation from the above gives: \(0 = 2qd + 482 - (pd + 112)\), which simplifies to \(p - 2q = \frac{370}{d}\). Since \(p - 2q\) is an integer, \(\frac{370}{d}\) must also be an integer, implying that \(d\) is a factor of 370. However, given that \(d > 241\), it can only be 370 itself.
Answer: D
19 Nov 2023, 20:36
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Bunuel
I was running out of time while solving this sum. What I started plugging in values.
We know d>241.
Lets assume d= 258 & x=241, doesn't help
d=353, doesn't help
d=370, Fits perfectly! (241/370 & 482/370)
That's it!
