Bunuel wrote:
Official Solution:
If x is an integer, how many even numbers does set \(\{0, x, x^2, x^3, ..., x^9\}\) contain?
We have the set with 10 terms: \(\{0, x, x^2, x^3, ..., x^9\}\).
Note that if \(x=odd\) then the set will contain one even (0) and 9 odd terms (as if \(x=odd\), then \(x^2=odd\), \(x^3=odd\), ..., \(x^9=odd\)) and if \(x=even\) then the set will contain all even terms (as if \(x=even\), then \(x^2=even\), \(x^3=even\), ..., \(x^9=even\)).
Also note that, the standard deviation is always more than or equal to zero: \(SD \ge 0\). SD is 0 only when the list contains all identical elements (or which is same only 1 element).
(1) The mean of the set is even. Since \(mean=\frac{sum}{10}=even\), then \(sum=10*even=even\). So, we have that \(0+x+x^2+x^3+...+x^9=even\) or \(x+x^2+x^3+...+x^9=even\), which implies that \(x=even\) (if \(x=odd\) then the sum of 9 odd numbers would be odd). \(x=even\) means that all 10 terms in the set are even. Sufficient.
(2) The standard deviation of the set is 0. According to the above, all 10 terms of the set are identical and since the first term is 0, then all other terms must equal to zero, hence all 10 terms in the set are even. Sufficient.
Answer: D
Hi Bunuel,
Are there any rules that support this statement "if \(x=odd\) then the sum of 9 odd numbers would be odd" ? We know the basic rules of addition, e.g. O+O=E etc. but I can't find any rule regarding the sum of an odd number (>1) of odd integers. Is that correct to say that:
- sum of an even number of even integers is even
- sum of an even number of odd integers is even
- sum of an odd number of even integers is even
- sum of an odd number of odd integers is odd
Thanks!