Question

Which is correct regarding the statement: “If x is an odd integer, then the median of x , x + 2, x + 6, and x + 10 is an odd number”

A.) the statement is sometimes true


B.) the statement is always false


C.) the statement is always true


D.) there is not enough information provided to answer the question

Answer 1

x is an odd number

x+2 is also odd (consider the example x = 3 so x+2 = 3+2 = 5). Adding 2 to any odd number is always odd

Similarly, so is x+6 (since we have x+2+2+2).

And so is x+10 (x+2+2+2+2+2).

So every value in this list is an odd number. The middle most values are x+2 and x+6 which are both odd.

Adding any two odd numbers together yields an even number. For example 3+5 = 8. Divide this even number in half and we may or may not get an odd number (eg: 8/2 = 4 and 6/2 = 3)

So this statement is sometimes true

Answer 2

any odd number can be represented by 2n+1 where n is an integer

the median will be the middle number, if there are 2, then it is the average of the 2 middle numbers when they are arranged in increasing order

we can see that there are 4 numbers, the middle 2 are x+2 and x+6
the average of those is (2x+8)/2=x+4

if x is odd, then we can replace x with 2n+1
2n+1+4=2n+4+1=2(n+4)+1, and n+4 is an integer so that is an odd result, bein 2(n+4)+1 is the same as 2m+1 where m is an integer

the statement is always true

C is answer