Subjective Type

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

Solution

By Euclid's division algorithm , $$ a = bq + r $$ where $$ a , b , q , r $$ are non-negative integers and $$ 0 \leq r < b$$.
On putting $$ b = 4 $$ we get
$$ a = 4q + r $$ ....(i)

When $$ r = 0 , a = 4q $$
Squaring both sides, we get
$$a^2= (4q )^2 $$
$$ = 4(4q^2 ) $$
$$ = 4m $$ is perfect square for $$ m = 4q^2$$

When $$ r = 1 , a = 4q + 1 $$
Squaring both sides, we get
$$a^2= (4q + 1)^2 $$
$$ = (4q)^2 + (1)^2 + 2(4q) $$
$$ = 4(4q^2 + 2q ) + 1 $$
$$ = 4m + 1 $$ is perfect square for $$ m = 4q^2 + 2q $$

When $$ r = 2 , a = 4q + 2 $$
Squaring both sides, we get
$$a^2= (4q + 2)^2 $$
$$ = (4q)^2 + (2)^2 + 2(4q)(2) $$
$$ = 4(4q^2 + 4q + 1 )$$
$$ = 4m $$ is perfect square for $$ m = 4q^2 + 4q +1$$


When $$ r = 3 , a = 4q + 3 $$
Squaring both sides, we get
$$ a^2=(4q + 3)^2 $$
$$= (4q)^2 + (3)^2 + 2(4q) (3) $$
$$ = 16q^2 + 9 + 24q $$
$$ = 16q^2 + 24q + 8 + 1 $$
$$ = 4(4q^2 \,6q + 2) + 1 $$
$$ = 4m + 1 $$ is perfect square for some value of $$m$$.

Hence, the square on any integer is either of the form $$4q$$ or $$(4q+1)$$ for some integer $$q$$.

SIMILAR QUESTIONS

Number Systems

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

Number Systems

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

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