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07 Apr 2021, 10:00
Kudos
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
37% (02:20) correct
63%
(02:24)
wrong
based on 102
sessions
History
Date
Time
Result
Not Attempted Yet
If x is a prime number greater than 5, y is a positive integer, and \(5y = x^2 + x\), then y must be divisible by which of the following?
I. 5
II. 2x
III. x + 1
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
I. 5
II. 2x
III. x + 1
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
14 Apr 2021, 21:00
Kudos
Bookmarks
Bunuel
You can solve this question even not analyzing option II at all.
Since \(5y=x(x+1)\), then \(x+1\) must be a multiple of 5, thus the units digit of \(x\) must be 9 (it cannot be 4, because in this case \(x\) would be even and thus not a prime number).
Now, if \(x=19\), then neither I nor III is true. The only option that does not contains I or III is B. So, B must be correct.
Answer: B.
If you interested why II must be true consider this: since \(x\) is a prime number greater than 5, then it's odd --> \(5y=x(x+1)=odd(odd+1)=odd*even=even\) --> \(5y=even\) --> \(y=even\).
Also, \(\frac{y}{x}=\frac{x+1}{5}\). we know that \(x+1\) is a multiple of 5, thus \(\frac{y}{x}=\frac{x+1}{5}=integer\) --> \(y\) is a multiple of \(x\). And since \(y\) is even, then \(y\) is a multiple of \(2x\).
Hope it helps.
General Discussion
Asked: If x is a prime number greater than 5, y is a positive integer, and \(5y = x^2 + x\), then y must be divisible by which of the following?
y = x(x+1)/5
y is divisible by 2 since x is a prime number greater than 5 (odd) and (x+1) is even
y is also divisible by x but may not be divisible by (x+1) if it is a multiple of 5.
For example, if x=19; x+1=20 but y = 19*20/5 = 76 is divisible by 2x = 38 but not divisible by (x+1) = 20
I. 5
II. 2x
III. x + 1
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
IMO B
y = x(x+1)/5
y is divisible by 2 since x is a prime number greater than 5 (odd) and (x+1) is even
y is also divisible by x but may not be divisible by (x+1) if it is a multiple of 5.
For example, if x=19; x+1=20 but y = 19*20/5 = 76 is divisible by 2x = 38 but not divisible by (x+1) = 20
I. 5
II. 2x
III. x + 1
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
IMO B
