Single Choice

$$n^2 -1$$ is divisible by $$8$$ , if n is

Aan integer
Correct Answer
Ba natural number
Can odd integer
Dan even integer

Solution

Given,

Any odd positive integer $$n$$ can be written in form of $$4q + 1$$ or $$4q + 3$$.

If $$n = 4q + 1$$, when $$n^2 - 1 = (4q + 1)^2 - 1 = 16q^2 + 8q + 1 - 1 = 8q(2q + 1)$$ which is divisible by $$8.$$

If $$n = 4q + 3$$, when $$n^2 - 1 = (4q + 3)^2 - 1 = 16q^2 + 24q + 9 - 1 = 8(2q^2 + 3q + 1)$$ which is divisible by $$8$$.

So, it is clear that $$n^2 - 1$$ is divisible by $$8$$, if $$n$$ is an odd positive integer.

SIMILAR QUESTIONS

Number Systems

If $$n$$ is an odd integer then show that $${n^2} - 1$$ is divisible by 8

Number Systems

Show that the square of any integer is either of the form $$ 4q$$ or $$4q+1$$ for some integer $$q.$$

Number Systems

A positive integer is of the form $$3q+1, q$$ being a natural number. Can you write its square in any form other than $$3m+1$$ i.e., $$3m$$ or $$3m+2$$ for some integer $$m$$? Justify your answer.

Number Systems

Write whether the square of any positive integer can be of the form $$3m+2$$, where $$m$$ is a natural number. Justify your answer.

Number Systems

Show that the square of any positive integer cannot be of the form $$ (6m + 2) $$ or $$ (6m + 5) $$ for any integer $$ m $$ .

Number Systems

Show that the square on any odd integer is of the form $$ (4q + 1) $$ for some integer $$ q $$.

Number Systems

Show that the cube of a positive integer of the form $$ (6q + r) $$ where $$ q $$ is an integer and $$ r = 0 , 1 , 2, 3 , 4 $$ and $$ 5 $$ is also of the form $$ (6m + r) $$.

Number Systems

State Euclid's division lemma and give some examples.

Number Systems

Write whether the square of any positive integer can be of the form $$3m+2$$, where $$m$$ is a natural number. Justify your answer.

Number Systems

Use Euclid's algorithm to find the HCF of $$4052$$ and $$12576$$.

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