I dont understand this solution from MGMAT :(

Question

Hey guys and gals,

question: Is X^2 odd?

statement 2: The sum of x consecutive integers is divisible by x

the solution states that the sum divided by x is an integer and moreover the sum of x consecutive integers divided by x is the average of that set of x integers. this therefore proves that x^2 is odd....how does that make sense?

anyone care to explain?

C210

C210

siddharthmuzumdar · Answer

x^2 will be odd only when x is odd. Thus, you are basically asked whether x is odd.

Now, the given statement says that the sum of x consecutive integers is an integer divisible by x.

To interpret this, you have to know a basic rule:

For all positive integers
The sum of n consecutive integers divided by n will be an integer if n is odd. 
The sum of n consecutive integers divided by n will NOT be an integer if n is even.

To verify, let us check.
For n = 2, take any 2 consecutive integers and find their average. The result will NEVER be an integer. The same applies for n = 4 and all even numbers.
For n = 3, take any 3 consecutive integers and find their average. The result will ALWAYS be an integer. The same applies for n = 5 and all odd numbers.

Thus, the given statement tells us that x is odd. Thus x^2 is odd.

Hope that helps.

c210 · Answer

ooooh! thanks senior manager!

c210

siddharthmuzumdar · Answer

I hope you meant my name. That's Siddharth, not Senior Manager. :D

Anyway, glad it helped. Consider KUDOS if you think it's worth the effort.

Bunuel · Answer

Hi there, When posting a question in PS/DS subforums make sure that the question is complete. So, please edit the question you've posted and add statement (1). As for your query, below are some tips about consecutive integers (for more on this topic see Number Theory Chapter of Math Book: math-number-theory-88376.html). Consecutive Integers Consecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers. • Sum of n consecutive integers equals the mean multiplied by the number of terms, n . Given consecutive integers \{-3, -2, -1, 0, 1,2\} , mean=\frac{-3+2}{2}=-\frac{1}{2} , (mean equals to the average of the first and last terms), so the sum equals to -\frac{1}{2}*6=-3 . • If n is odd, the sum of consecutive integers is always divisible by n. Given \{9,10,11\} , we have n=3 consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3. • If n is even, the sum of consecutive integers is never divisible by n. Given \{9,10,11,12\} , we have n=4 consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4. • The product of n consecutive integers is always divisible by n! . Given n=4 consecutive integers: \{3,4,5,6\} . The product of 3*4*5*6 is 360, which is divisible by 4!=24. So, if statement (2) says that "the sum of x consecutive integers is divisible by x", according to the second point above x is odd, which makes x^2 an odd number. As for your second question: A set of consecutive integers is an evenly spaced set (AP). • In any evenly spaced set the arithmetic mean (average) is equal to the median . • The sum of the elements in any evenly spaced set equals to the mean multiplied by the number of terms: {sum}={mean}*{# of terms} Thus, for original question: sum/x must equal to the mean (and median) and as median of odd consecutive integers is an integer itself, then sum/x=mean=median=integer. Hope it's clear.

I dont understand this solution from MGMAT :(

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