Question

using euclids division lemma show that the square of any positive integer is of the 5m, 5m+1 or 5m+4 for some integer m

Answer 1

Heya!!

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Solution:-
Let 'a' be any positive integer and b = 5.
Applying Euclid's division lemma with a and b,we get,

a =5q+r,where 0<=r<5 and q is some integer.

Therefore, r =0,1,2,3,4

When ,r =0,1,2,3,4,then,
a =5q,5q+1,5q+2,5q+3,5q+4.

For some integer m ,square of positive integers is given by:
Now,When a =5q
=>a^2 = (5q)^2
= 25q^2
= 5×5q^2
= 5m , where m=5q^2.
When, a = 5q+1
=>a^2 = (5q+1)^2
=25q^2+10q+1
=5(5q^2+2q)+1
=5m+1, where m =(5q^2+2q)
Similarly, Do upto a =5q +4

Therefore, Square of any positive integer is of the form 5m,5m+1 or 5m+4,for some integer m.


Hope it helps you.

Mark as Brainliest answer.

Answer 2

Step-by-step explanation:

Question : -

→ Use Euclid's Division lemma to show that the Square of any positive integer cannot be of form 5m + 2 or 5m + 3 for some integer m.

$\huge \pink{ \mid{ \underline{ \overline{ \tt Answer: -}} \mid}}$

▶ Step-by-step explanation : -

Let ‘a’ be the any positive integer .

And, b = 5 .

→ Using Euclid's division lemma :-

==> a = bq + r ; 0 ≤ r < b .

==> 0 ≤ r < 5 .

•°• Possible values of r = 0, 1, 2, 3, 4 .

→ Taking r = 0 .

Then, a = bq + r .

==> a = 5q + 0 .

==> a = ( 5q )² .

==> a = 5( 5q² ) .

•°• a = 5m . [ Where m = 5q² ] .

→ Taking r = 1 .

==> a = 5q + 1 .

==> a = ( 5q + 1 )² .

==> a = 25q² + 10q + 1 .

==> a = 5( 5q² + 2q ) + 1 .

•°• a = 5m + 1 . [ Where m = 5q² + 2q ] .

→ Taking r = 2 .

==> a = 5q + 2 .

==> a = ( 5q + 2 )² .

==> a = 25q² + 20q + 4 .

==> a = 5( 5q² + 4q ) + 4 .

•°• a = 5m + 4 . [ Where m = 5q² + 4q ] .

→ Taking r = 3 .

==> a = 5q + 3 .

==> a = ( 5q + 3 )² .

==> a = 25q² + 30q + 9 .

==> a = 25q² + 30q + 5 + 4 .

==> a = 5( 5q² + 6q + 1 ) + 4 .

•°• a = 5m + 4 . [ Where m = 5q² + 6q + 1 ] .

→ Taking r = 4 .

==> a = 5q + 4 .

==> a = ( 5q + 4 )² .

==> a = 25q² + 40q + 16 .

==> a = 25q² + 40q + 15 + 1 .

==> a = 5( 5q² + 8q + 3 ) + 1 .

•°• a = 5m + 1 . [ Where m = 5q² + 8q + 3 ] .

→ Therefore, square of any positive integer in cannot be of the form 5m + 2 or 5m + 3 .

✔✔ Hence, it is proved ✅✅.

$\huge \orange{ \boxed{ \boxed{ \mathscr{THANKS}}}}$