Data Sufficiency (DS) | Re: If f(n) denotes the remainder of n^2 divided by k, where n and k are

Sol:IMO-Option-E


\(f(n)\) denotes the remainder of\(n^2\) divided by k == >\(n^2\) = kQ +\(f(n)\) , where Quotient, Q > 0
or\(f(n)\) =\(n^2\) -kQ

St-1:
\( f(k+12) =16\)
==> 16 = (k+12)^2 - kQ
L.H.S = even
Depending whether k & Q are odd or even four cases arise:
Case-1:
k= Odd , Q = Odd
R.H.S = Odd - Odd*Odd = Even = L.H.S ---> Valid Case
Case-2:
k = Odd , Q=Even
R.H.S = Odd - Odd*Even= Odd ---> Invalid Case
Case-3:
k= Even , Q = Odd
R.H.S = Even- Even*Odd = Odd ---> Invalid Case
Case-4:
k = Even; Q=Even
R.H.S = Even - Even*Even= Odd = L.H.S ---> Valid Case
We have two valid cases, in first k is odd and in another k is even. Statement isInsufficient

St-2:
\( f(k+11) =9\)
==> 9 = (k+11)^2 - kQ
L.H.S = Odd
Depending whether k & Q are odd or even four cases arise:
Case-1:
k= Odd , Q = Odd
R.H.S = Even - Odd*Odd = Odd = L.H.S --->
...

Statistics : Posted by Aryaa03 • on 15 Jan 2024, 01:30 • Replies 1 • Views 128

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