Is n 2 1 divisible by 8?

4Q² + 4Q = 4(2)² + 4(2) =16 + 8 = 24, it is also divisible by 8 . It is concluded that 4Q² + 4Q is divisible by 8 for all natural numbers. Hence, n² -1 is divisible by 8 for all odd values of n.

How do you prove divisibility using mathematical induction?

Mathematical Induction for Divisibility

1. Show the basis step is true. That is, the statement is true for n=1.
2. Assume the statement is true for n=k. This step is called the induction hypothesis.
3. Prove the statement is true for n=k+1. This step is called the induction step.

What would be the value of N for which N 2 is divisible by 8?

Any odd positive integer is in the form of 4p + 1 or 4p+ 3 for some integer p. ⇒ (n2 – 1) is divisible by 8. ⇒ n2– 1 is divisible by 8. Therefore, n2– 1 is divisible by 8 if n is an odd positive integer.

What would be the value of n if’n square minus 1 is divisible by 8?

n^2 – 1 is divisible by 8 , if n is number.

How do you show that something is divisible by 8?

Divisibility by 8

1. Rule for Divisibility by 8. A number with at least 3 digits is divisible by 8 if its last three digits form a number divisible by 8.
2. Examples. A.)
3. Proof. For any integer x written as anan-1an-2…a2a1a0, we will show that x is divisible by 8 if a2a1a0 is divisible by 8.

How do you prove that n^2 -1 is divisible by 8?

Show that n^2 -1 is divisible by 8, if n is an odd positive integer. n=2k+1 where k is a non-negative integer. Since in both cases, 8 divides n²-1, therefore it is proved that 8 divides n²-1 in for all positive values of n.

What is an example of a direct proof of an odd integer?

This is a nice example of a direct proof. You start with the facts that if ϕ is your positive odd integer, then it is in the form ϕ = 2n + 1 where n is an integer and ϕ2 − 1 = 8p (p ∈ Z) is true. Recall that if a number is divisible by 8, then 8 is one of its factors. This is something like a “case-by-case” proof.

Why are there no numbers that are divisible by 3?

The simplest explanation follows from Modular Arithmetic. Any integer must be either 0, 1, or 2 (mod 3). n2 + 1 sends these to 1, 2, and 2 (mod 3). Since none are 0 (mod3), none are divisible by 3.

What are the rules for divisibility?

Divisibility Rules for some Selected Integers Divisibility by 1: Every number is divisible by \\(1\\). Divisibility by 2: The number should have \\(0, \\ 2, \\ 4, \\ 6,\\) or \\(8\\) as the units digit. Divisibility by 3: The sum of digits of the number must be divisible by \\(3\\).