Question

Class 10 Maths !!!!!!!!
Only For Intelligents..........

A positive integer is of the form 3q+1 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.

Answer 1

Solution:-

Let "a" be any positive Integer and "b = 3".

By Euclid Division Lemma,

The possible value of "r" can be 0, 1 and 2.

=) "a" can take values of 3q + 0 , 3q + 1 and 3q + 2.

Case |,

a = 3q + 0

Squaring on both the sides.

=) a² = ( 3q + 0)²

=) a² = 9q²

=) a² = 3 ( 3q²)

Taking ( a² = x) and (3q² = m)

=) x = 3m

Case ||,

a = 3q + 1

Squaring on both the sides.

=) a² = ( 3q + 1)²

=) a² = 9q² + 1 + 6q

=) a² = 3( 3q² + 2q) + 1

Taking ( a² = x) and ( 3q² + 2q = m)

=) x = 3m + 1.

Case |||,

a = 3q + 2

Squaring on both the sides.

=) a² = ( 3q +2)²

=) a² = 9q² + 4 + 12q

=) a² = 9q² + 12q + 3 + 1

=) a² = 3( 3q² + 4q + 1) + 1

Taking ( a² = x) and ( 3q² + 4q +1 = m).

=) a² = 3m + 1.

From Case (1), (2) and (3).

Sum of y Positive Integer is of the form either (3m) or (3m+1).

Answer 2

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Note: the numbers after variables are their powers.

It is necessary to solve all the values of r in the exam

Answer:

By using Euclid's Division Lemma, a=bq+r

Where, 0 ≤ r < b here, b=3 therefore, r= 0,1 or 2

So,

1. r= 0

2. r= 1

(skipping to r= 2 NOT TO BE DONE IN EXAM)

3. r= 2

a²= (3q+2)²

a²= 9q² + 12q + 4

a²= 3(3q² + 4q) + 4

Now, let (3q2 + 4q) be m

Therefore, a²= 3m + 4