State true or false:The square of any positive integer is of the form 5q, 5q+1 and 5q+4 for some integer q.

Question

State true or false:The square of any positive integer is of the form 5q, 5q+1 and 5q+4 for some integer q.

Accepted Answer

We have to check whether the square of any positive integer is of the form

5 q, 5 q + 1 and 5 q + 4

for some integer q.
Euclid's Division Lemma states that, if two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition:

a = bq + r

where

0 \leq r \leq b

Consider ‘n’ as a positive integer for some integer q.

n = bq + r

From Euclid's Division Lemma for b = 5,

n = 5 k + r

where

0 \leq r \leq 5

…... (1)
Substitute

r = 0

(1) and square on both sides,

n^{2} = {(5 k)}^{2}

\Rightarrow n^{2} = 5 (5 k^{2})

Let

5 k^{2}

be ‘q’, then

n^{2} = 5 q

.
Here, q is some integer.
Substitute

r = 1

in (1) and square on both sides,

n^{2} = {(5 k + 1)}^{2}

\Rightarrow n^{2} = 25 k^{2} + 10 k + 1

\Rightarrow n^{2} = 5 (5 k^{2} + 2 k) + 1

Let

5 k^{2} + 2 k

be ‘q’, then

n^{2} = 5 q + 1

.
Here, q is some integer.
Substitute

r = 2

in (1) and square on both sides,

n^{2} = {(5 k + 3)}^{2}

\Rightarrow n^{2} = {25 k}^{2} + 20 k + 4

\Rightarrow n^{2} = {5 (5 k}^{2} + 4 k) + 4

Let

(5 k^{2} + 4 k)

be ‘q’.

\Rightarrow n^{2} = 5 q + 4

Here, q is some integer.
Substitute

r = 3

in (1) and square on both sides,

n^{2} = {(5 k + 3)}^{2}

\Rightarrow n^{2} = {25 k}^{2} + 30 k + 9

\Rightarrow n^{2} = {5 (5 k}^{2} + 6 k + 1) + 4

Let

(5 k^{2} + 6 k + 1)

be ‘q’.

\Rightarrow n^{2} = 5 q + 4

Here, q is some integer.
Substitute

r = 4

in (1) and square on both sides,

n^{2} = {(5 k + 4)}^{2}

\Rightarrow n^{2} = {25 k}^{2} + 40 k + 16

\Rightarrow n^{2} = {5 (5 k}^{2} + 8 k + 3) + 1

Let

(5 k^{2} + 8 k + 3)

be ‘q’.

\Rightarrow n^{2} = 5 q + 1

Here, q is some integer.
Therefore, it is true that the square of any positive integer is of the form

5 q, 5 q + 1 and 5 q + 4

for some integer q.
Hence, option 1 is correct.

State true or false:
The square of any positive integer is of the form $5 q, 5 q + 1 and 5 q + 4$ for some integer q.

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