ACT questions again?

Question

Thanks to everyone who can help me. Please, if you can, provide me with a step by step explanation.

1. Which of the following is true for all consecutive intergers m and n such that m is less than n?
a. m is odd
b. n is odd
c. n-m is odd
d. n squared - m squared is odd
e. m squared + n squared is even

2. Two numbers are reciprocals if their product is equal to 1. If x and y are reciprocals and x is greater than 1, then y must be:

Answer 1

1) as stated, the two integers m and n are consecutive wherein m
a)m can be odd but not all the time for it could also be even since they are consecutive and even and odd follow each other. Thus, it is NOT ALWAYS TRUE.

b)n could be odd but not all the time for it could also be even. this shares the same reason with a). thus, it is NOT ALWAYS TRUE.

c)we know that m and n are consecutive. thus, we can represent m as n-1 or m=n-1. So, n-m= n-(n-1) = n-n+1. This simplifies to 1. Since 1 is odd, then IT IS ALWAYS TRUE. THe simple explanation for this is that the difference of two consecutive numbers is always 1, and 1 is odd.

d)Recall that n^2-m^2 = (n-m)(n+m). since n and m are consecutive, n-m=1, n^2-m^2= (n+m). Also, we know that one of m and n is even and the other is odd. Also, odd+even is always odd. Thus n^2 - m^2 is odd is ALWAYS TRUE.

e) We know that one of m and n is odd or even. Thus, m^2+n^2 = an odd# ^2 + even# ^2. we also know that odd^2 is always odd and even^2 is always even. odd+even as we recall is always odd, thus m^2+n^2 is even is ALWAYS FALSE.

2)we can represent x and y into an equation: xy=1 since both of them are reciprocals. If x is greater than 1, then y must be a fraction. Take this as an example: if x=2 which is greater than 1, then y=1/2 to have a product equal to 1.

hope its clear enough! have a good day:D!

Answer 2

2. y is less than 1
No step by step, just look at some examples. The reciprocal of 2 is 1/2, which is less than 1.

1. n = m + 1, since they are consecutive integers. If you subtract n - m, you get m + 1 - m , which is 1, which is odd, so C is correct.

Answer 3

1. All but e can be true. c and d are always true.
2. y < 1