Question

1) Use Euclid's division lemma to show that the square of any positive integer is either of
the form 3m or 3m + 1 for some integer m.
[Hint: Let x be any positive integer then it is of the form 3q, 3q+1 or 3q + 2. Now square
each of these and show that they can be rewritten in the form 3m or 3m +1.]

2) Use Euclid's division lemma to show that the cube of any positive integer is of the form
9m, 9m + 1 or 9m +8.


NCERT MATHS BOOK CLASS 10 CH 1 EXERCISE 1.PLS DO EXPLAIN ME IN DETAIL​

Answer 1

Answer:

Step-by-step explanation:

Let 'a' be any positive integer.

On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.

Such that ,

a = 3q + r , where r = 0 ,1 , 2

When, r = 0

∴ a = 3q

When, r = 1

∴ a = 3q + 1

When, r = 2

∴ a = 3q + 2

When , a = 3q

On squaring both the sides,

When, a = 3q + 1

On squaring both the sides ,

When, a = 3q + 2

On squaring both the sides,

Therefore , the square of any positive integer is either of the form 3m or 3m+1.