Write whether the square of any positive integer can be of the form $3m + 2$, where $m$ is a natural number. Justify your answer.
Given:
"Square of any positive integer cannot be of the form $3m+2$, where $m$ is a natural number".
To do:
We have to find whether the given statement is true or false.
Solution:
According to Euclid's lemma,
If $a$ and $b$ are two positive integers;
- $a\ =\ bq\ +\ r$, where $0\ \underline{< }\ r\ <\ b$.
If $b\ =\ 3$, then;
- $a\ =\ 3q\ +\ r$, where $0\ \underline{< }\ r\ <\ 3$.
- So, $r\ =\ 0,\ 1,\ 2$
When, $r\ =\ 0$:
$a\ =\ 3q$
Squaring on both sides, we get:
$a^2\ = (3q)^2$
$a^2\ = 9q^2$
$a^2\ = 3(3q^2)$
$a^2\ = 3m$, where $m\ =\ 3q^2$
When, $r\ =\ 1$:
$a\ =\ 3q\ +\ 1$
Squaring on both sides, we get:
$a^2\ = (3q\ +\ 1)^2$
$a^2\ = 9q^2\ +\ 6q\ + 1$
$a^2\ = 3(3q^2\ +\ 2q)\ +\ 1$
$a^2\ = 3m\ +\ 1$, where $m\ =\ 3q^2\ +\ 2q$
When, $r\ =\ 2$:
$a\ =\ 3q\ +\ 2$
Squaring on both sides, we get:
$a^2\ = (3q\ +\ 2)^2$
$a^2\ = 9q^2\ +\ 12q\ + 4$
$a^2\ = 9q^2\ +\ 12q\ + 3\ +\ 1$
$a^2\ = 3(3q^2\ +\ 4q\ +\ 1)\ +\ 1$
$a^2\ = 3m\ +\ 1$, where $m\ =\ 3q^2\ +\ 4q\ +\ 1$
Hence, the square of any positive number cannot be of the form $3m\ +\ 2$.
Related Articles
- Show that the square of any positive integer cannot be of the form $3m+2$, where $m$ is a natural number.
- A positive integer is of the form $3q+1$, $q$ being a natural number. Can you write its square in any form other than $3m+1$, $3m$ or $3m+2$ for some integer $m$? Justify your answer.
- 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.
- Write whether every positive integer can be of the form $4q + 2$, where $q$ is an integer. Justify your answer.
- Prove that the square of any positive integer is of the form $3m$ or, $3m+1$ but not of the form $3m+2$.
- Show that the square of any positive integer cannot be of the form $6m+2$ or $6m+5$ for any positive integer $m$.
- Show that the square of any positive integer cannot be of the form $6m+ 2$ or $6m + 5$ for any integer $m$.
- Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m$+$1 for some integer m.
- Show that the square of any positive integer cannot be of the form $5q + 2$ or $5q + 3$ for any integer $q$.
- Write 'True' or 'False' and justify your answer in each of the following:The value of \( 2 \sin \theta \) can be \( a+\frac{1}{a} \), where \( a \) is a positive number, and \( a \neq 1 \)
- Prove that the square of any positive integer of the form $5q+1$ is of the same form.
- Show that the square of any odd integer is of the form $4m + 1$, for some integer $m$.
- Show that square of any positive integer is of the form \( 5p, 5p+1, 5p+4 \).
- Prove that the square of any positive integer is of the form $4q$ or $4q+1$ for some integer q.
- State whether the following statements af true or false. Justify your answers.(i) Every irrational number is a real number.(ii) Every point on the number line is of the form \( \sqrt{m} \), where \( m \) is a natural number(iii) Every real number is an irrational number.
Kickstart Your Career
Get certified by completing the course
Get Started