DISCRETE MATHS

posted by .

We need to show that 4 divides 1-n2 whenever n is an odd positive integer.
If n is an odd positive integer then by definition
n = 2k+1 for some non negative integer, k.
Now 1 - n2 = 1 - (2k+1)2 = -4k2-4k = 4 (-k2-4k).
k is a nonnegative integer, hence -k2-4k is an integer. Thus by definition
of divisibility we conclude that 4 divides 1-n2.

Similar Questions

1. discrete math

Prove by contradiction that for any even integer a and any odd integer b, 4 does not divide (a^2 + 2b^2). Proposition: That 4k (k is any integer) = a^2 +2b^2, and a is even, and b is odd. But 4k is even (product of any integer and …
2. math

Show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is any positive integer
3. math

show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is any positive integer.
4. Discrete Math

Theorem: For every integer n, if x and y are positive integers with max(x, y) = n, then x = y. Basic Step: Suppose that n = 1. If max(x, y) = 1 and x and y are positive integers, we have x = 1 and y = 1. Inductive Step: Let k be a …
5. Algebra

The sum of two consecutive odd integers is 56. A. Define a variable for the smaller integer. B. What must you add to an odd integer to get the next greater odd integer?
6. maths

prove that any odd positive integer of 8q+1,where q is any integer?
7. DISCRETE MATHS

Prove that if n is an odd positive integer, then 1 ≡ n2 (mod 4).
8. algebra 1

PLEASE HELP! NO ONE ANSWERED THE QUESTION three times an odd integer is eleven less than four times the next greater even integer. What is the odd integer?
9. maths

the non- decreasing sequence of odd integers {a1, a2, a3, . . .} = {1,3,3,3,5,5,5,5,5,...} each positive odd integer k appears k times. it is a fact that there are integers b, c, and d such that, for all positive integers n, aĆ± = …
10. Algebra 1 Polynomials

Suppose n is an integer. Select all statements below that are true: (choose 3) A) n^2 + n is always an even integer*** B) n^2 + n is always an even integer when n is even*** C) n^2 + n is always an even integer when n is odd*** D) …

More Similar Questions