Subjective Type

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

Solution

Any odd positive integer is in the form of 4p+1 & 4p+3 for n=4p+1
$$(n^{2}-1)=(1p+1)^{2}-1=16p^{2}+8p+1=16p^{2}+8p=8p(2p+1)$$
$$=(n^{2}-1)$$ is divisible by 8.
$$(n^{2}-1)=(4p+3)^{2}-1=8(2p^{2}+3p+1)$$
$$\Rightarrow n^{2}-1$$ is divisible by 8
$$n^{2}-1$$ is divisible by 8

SIMILAR QUESTIONS

Number Systems

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

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 $$12576$$.

Contact Details