Question

Without using induction, prove that for x is an odd, positive integer, 3x ≡−1 (mod 4). I'm not sure how to approach the problem. I thought to assume that x=2a+1 and then show that 3^x +1 is divisible by 4 and thus congruent to 3x=-1(mod4) but I'm stuck.

Answer #1

Prove by induction that k ^(2) − 1 is divisible by 8 for every
positive odd integer k.

Prove that if x+ \frac{1}{x} is integer then x^n+ \frac{1}{x^n}
is also integer for any positive integer n.
KEY NOTE: PROVE BY INDUCTION

What is the smallest positive integer x satisfying x ≡
1 mod 9, x ≡ 6 mod 10, x ≡ 4 mod 11?

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

Suppose f(x)=x6+3x+1f(x)=x6+3x+1. In this problem, we will show
that ff has exactly one root (or zero) in the interval
[−4,−1][−4,−1].
(a) First, we show that f has a root in the interval
(−4,−1)(−4,−1). Since f is a SELECT ONE!!!! (continuous)
(differentiable) (polynomial) function on the
interval [−4,−1] and f(−4)= ____?!!!!!!!
the graph of y=f(x)y must cross the xx-axis at some point in the
interval (−4,−1) by the SELECT ONE!!!!!! (intermediate value
theorem) (mean value theorem) (squeeze theorem) (Rolle's theorem)
.Thus, ff...

1. For a pair of sample x- and y-values, what is the difference
between the observed value of y and the predicted value of y? a) An
outlier b) The explanatory variable c) A residual d) The response
variable
2. Which of the following statements is false:
a) The correlation coefficient is unitless. b) A correlation
coefficient of 0.62 suggests a stronger correlation than a
correlation coefficient of -0.82. c) The correlation coefficient,
r, is always between -1 and 1....

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