Question

show that square of any positive integer cannot be of the form 5q + 2 or 5q +3 for any integer q​

Answer 1

Answer:

Step-by-step explanation:

Let a be any positive integer

By Euclid's division lemma

a=bm+r

a=5m+r   (when b=5)

So, r can be 0,1,2,3,4  

Case 1:

∴a=5m   (when r=0)

a  

2

=25m  

2

 

a  

2

=5(5m  

2

)=5q

where q=5m  

2

 

Case 2:

when r=1

a=5m+1

a  

2

=(5m+1)  

2

=25m  

2

+10m+1

a  

2

=5(5m  

2

+2m)+1

=5q+1    where q=5m  

2

+2m

Case 3:

a=5m+2

a  

2

=25m  

2

+20m+4

a  

2

=5(5m  

2

+4m)+4

5q+4

where q=5m  

2

+4m

Case 4:

a=5m+3

a  

2

=25m  

2

+30m+9

=25m  

2

+30m+5+4

=5(5m  

2

+6m+1)+4

=5q+4

where q=5m  

2

+6m+1

Case 5:

a=5m+4

a  

2

=25m  

2

+40m+16=25m  

2

+40m+15+1

=5(5m  

2

+8m+3)+1

=5q+1       where q=5m  

2

+8m+3

From these cases, we see that square of any positive no can't be of the form 5q+2,5q+3.

Answer 2

Question:

Show that square of any positive integer cannot be of the form 5q + 2 or 5q +3 for any integer q.

Solution:

Let $x$ be any positive integer. When we divide $x$ by $\mathsf{5}$, the remainder is either 0 or 1 or 2 or 3 or 4. So, $x$ can be written as-: $x=5m$ or $x=5m+1$ or $x=5m+2$ or $x=5m+3$ or $x=5m+4$. Thus, we have the following cases:

Case ɪ When, $x=5m:$

ㅤㅤ$:\implies$ $\mathsf{x²=25m²}$

ㅤㅤ$\mathsf{x²=5(5m²)=5q\: where\: q=5m²}$

Case ɪɪ When, $x=5m+1:$

ㅤㅤ$:\implies$ $\mathsf{x²=(5m+1)²}$

ㅤㅤ$:\implies$ $\mathsf{x²=25m²+10m+1}$

ㅤㅤ$:\implies$ $\mathsf{x²=5(5m²+2m)+1}$

ㅤㅤ$\mathsf{x²=5q+1\: where\: q=5m²+2m}$

Case ɪɪɪ When $x=5m+2:$

ㅤㅤ$:\implies$ $\mathsf{x²=(5m+2)²}$

ㅤㅤ$:\implies$ $\mathsf{x²=25m²+20m+4}$

ㅤㅤ$:\implies$ $\mathsf{x²=5(5m²+4m)+4}$

ㅤㅤ$\mathsf{x²=5q+4\: where\: q=5m²+4m}$

Case ɪᴠ When $x=5m+3:$

ㅤㅤ$:\implies$ $\mathsf{x²=(5m+3)²}$

ㅤㅤ$:\implies$ $\mathsf{x²=25m²+30m+9}$

ㅤㅤ$:\implies$ $\mathsf{x²=(5m²+30m+5)+4}$

ㅤㅤ$:\implies$ $\mathsf{x²=5(5m²+6m+1)+4}$

ㅤㅤ$\mathsf{x²=5q+4\:where,q=5m²+6m+1}$

Case ᴠ When, $\mathsf{x=5m+4:}$

ㅤㅤ$:\implies$ $\mathsf{x²=(5m+4)²}$

ㅤㅤ$:\implies$ $\mathsf{x²=25m²+40m+16}$

ㅤㅤ$:\implies$ $\mathsf{x²=5(5m²+8m+3)+1}$

ㅤㅤ$\mathsf{x²=5q+1\:where,q=5m²+8m+3}$

Hence, $\mathsf{x}$ is of the form $5q$ or $5q+1$, $5q+4$. So it cannot be of the form $5q+2$ or $5q+3$.

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