Question

pls solve it's urgent ........

Answer 1

the answer for this question .

Answer 2

HELLO DOST

HERE IS YOUR ANSWER EXPLAINED

To Prove: Square of any number is of the form 3 m or 3 m +1

Proof: to prove this statement from Euclid's division lemma, take any number as a divisor, in question we have 3m and 3m + 1 as the form

So,

By taking, ’ a’ as any positive integer and b = 3.

Applying Euclid’s algorithm

a = 3q + r

Here, as the divisor is 3 there can be only 3 remainders, 0, 1 and 2

So, putting all the possible values of remainder in, a = 3q + r

a = 3q or 3q+1 or 3q+2

And now squaring all the values,

When a= 3q

Squaring both sides we get,

a2 = (3q)2

a2 = 9q2

a2 =3 (3q2)

a2 = 3 k1

Where k1 = 3q2

When a=3q+1

Squaring both sides we get,

a2 = (3q + 1)2

a2 = 9q2 + 6q + 1

a2 =3( 3q2 + 2q )+ 1

a2 = 3k2 + 1

Where k2 = 3q2 + 2q

When a = 3q+2

Squaring both sides we get,

a2 = (3q + 2)2

a2 = 9q2 + 12q + 4

a2 = 9q2 + 12q + 3+1

a2 = 3( 3q2 + 4q + 1) +1

a2 = 3k3 + 1

Where k3 = 3q2 + 4q + 1

Where k1, k2 and k3 are some positive integers

Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.

HOPE IT HELPS YOU DEAR FRIEND

THANKS