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13 May 2020, 06:55
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Difficulty:
95%
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Question Stats:
41% (03:02) correct
59%
(02:37)
wrong
based on 68
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The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?
(1) 3n +2a is not divisible by 2
(2) 3a +2n is divisible by 2
(1) 3n +2a is not divisible by 2
(2) 3a +2n is divisible by 2
13 May 2020, 09:43
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SCI(a,n) = a+a+1+a+2+......+a+n-1= [a+a+a+...n times]+ [1+2+3+....+n-1]
SCI(a,n) \(= na +\frac{n*(n-1)}{2}\)
SCI(a,n)=\( \frac{n}{2}(2a+(n-1))\)
If n is odd, then (2a+n-1) is even. hence, \(\frac{n}{2}(2a+(n-1))\) is multiple of n
If n is even, then (2a+n-1) is odd. hence, \(\frac{n}{2}(2a+(n-1))\) is not a multiple of n.
Basically question stem is whether we can tell n is odd or even.
Statement 1-
3n+Even = Odd
3n = odd-even = odd
n must be odd
Sufficient
Statement 2-
3a+Even = Even
a= Even
We don't know anything about n
Insufficient
SCI(a,n) \(= na +\frac{n*(n-1)}{2}\)
SCI(a,n)=\( \frac{n}{2}(2a+(n-1))\)
If n is odd, then (2a+n-1) is even. hence, \(\frac{n}{2}(2a+(n-1))\) is multiple of n
If n is even, then (2a+n-1) is odd. hence, \(\frac{n}{2}(2a+(n-1))\) is not a multiple of n.
Basically question stem is whether we can tell n is odd or even.
Statement 1-
3n+Even = Odd
3n = odd-even = odd
n must be odd
Sufficient
Statement 2-
3a+Even = Even
a= Even
We don't know anything about n
Insufficient
Bunuel
General Discussion
13 May 2020, 09:52
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The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?
SCI(a, n) = a+ (a+1)+(a+2)+...+(a+n -1)
= n/2*[2a+n-1]
now,SCI(a, n) = an integer ( sum of the integers)
say,SCI(a, n) is divisible by n ,
SCI(a, n) /n = [2a+n-1]/2
so,[2a+n-1] must be divisible by 2
or,n-1 must be divisible by 2
or, n-1 must be even
or, n must be odd
so, the question asks IS N ODD ?
ST(1) 3n +2a is not divisible by 2
As 3n + 2a is not divisible by 2, 3n + 2a is odd
3n +2a = odd
3n + even = odd
3n = odd
n = odd sufficient
ST(2) 3a +2n is divisible by 2
As 3a + 2n is divisible by 2, 3a + 2n is even
3a + 2n = even
3a + even = even
3a = even
n may be even or may be odd insufficient
correct answer A
NOTES : check and analysis these examples
1+2+3 = 3/2*[1+3]
now, (1+2+3 )/3 = 3*[1+3] /2*3 = [1+3]/2 = integer so, divisible by 3 (n= ODD)
BUT,
1+2+3+4= 4/2*[1+4]
now, (1+2+3+4 )/4 = 4*[1+4] /2*4 = [1+4]/2 = NOT integer so, NOT divisible by 4 (n = EVEN)
in general ,
S=1+2+3+...+n = n(n+1)/2
if, we divide S by n , Savg = INTEGER if n= ODD
Savg = FRACTION if n= EVEN
for the same reason,
1. AVG OF TWO CONSECUTIVE INTEGERS ARE NOT DIVISIBLE BY TWO (fractional part is there)
2. MEDIAN of (=mean) of consecutive integers in a set having EVEN NUMBER of elements is always the avg of middle two terms.
SCI(a, n) = a+ (a+1)+(a+2)+...+(a+n -1)
= n/2*[2a+n-1]
now,SCI(a, n) = an integer ( sum of the integers)
say,SCI(a, n) is divisible by n ,
SCI(a, n) /n = [2a+n-1]/2
so,[2a+n-1] must be divisible by 2
or,n-1 must be divisible by 2
or, n-1 must be even
or, n must be odd
so, the question asks IS N ODD ?
ST(1) 3n +2a is not divisible by 2
As 3n + 2a is not divisible by 2, 3n + 2a is odd
3n +2a = odd
3n + even = odd
3n = odd
n = odd sufficient
ST(2) 3a +2n is divisible by 2
As 3a + 2n is divisible by 2, 3a + 2n is even
3a + 2n = even
3a + even = even
3a = even
n may be even or may be odd insufficient
correct answer A
NOTES : check and analysis these examples
1+2+3 = 3/2*[1+3]
now, (1+2+3 )/3 = 3*[1+3] /2*3 = [1+3]/2 = integer so, divisible by 3 (n= ODD)
BUT,
1+2+3+4= 4/2*[1+4]
now, (1+2+3+4 )/4 = 4*[1+4] /2*4 = [1+4]/2 = NOT integer so, NOT divisible by 4 (n = EVEN)
in general ,
S=1+2+3+...+n = n(n+1)/2
if, we divide S by n , Savg = INTEGER if n= ODD
Savg = FRACTION if n= EVEN
for the same reason,
1. AVG OF TWO CONSECUTIVE INTEGERS ARE NOT DIVISIBLE BY TWO (fractional part is there)
2. MEDIAN of (=mean) of consecutive integers in a set having EVEN NUMBER of elements is always the avg of middle two terms.
