Is K^2 odd? 1) K-1 is divisable by 2 2) The sum of K

When we say that number \(a\) is divisible by number \(b\) it means: \(a\) and \(b\) are integers AND \(\frac{a}{b}=integer\).

Is K^2 odd?

(1) K-1 is divisable by 2 --> \(k-1=2n\) --> \(k=2n+1=odd\) --> \(k^2=odd^2=odd\). Sufficient.

(2) The sum of K consecutive integer is divisible by K:
• If k is odd, the sum of k consecutive integers is always divisible by k. Given \(\{9,10,11\}\), we have \(k=3=odd\) consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If k is even, the sum of k consecutive integers is never divisible by k. Given \(\{9,10,11,12\}\), we have \(k=4=even\) consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

So as \(\frac{sum \ of \ k \ consecutive \ integers}{k}=integer\), then \(k=odd\) --> \(k^2=odd^2=odd\). Sufficient.

Answer: D.