1.1 Question 4,3,5

NCERT Solutions for Class 10 Maths Exercise 1.1

3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Ans. We have to find the HCF (616, 32) to find the maximum number of columns in which they can march.

To find the HCF, we can use Euclid’s algorithm.

The HCF (616, 32) is 8.

Therefore, they can march in 8 columns each.

NCERT Solutions for Class 10 Maths Exercise 1.1

4. Use Euclid’s division lemma to show that the square of any positive integer is either of 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.]

Ans. Let a be any positive integer and b = 3.

Then a = 3q + r for some integer q ≥ 0

And r = 0, 1, 2 because 0 ≤ r < 3

Therefore, a = 3q or 3q + 1 or 3q + 2

Or,

Where  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.

NCERT Solutions for Class 10 Maths Exercise 1.1

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

Ans. Let a be any positive integer and b = 3

a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

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

Therefore, every number can be represented as these three forms.

We have three cases.

Case 1: When a = 3q,

Where m is an integer such that m =3q³

Case 2: When a = 3q + 1,

Where m is an integer such that 

Case 3: When a = 3q + 2,

Where m is an integer such that 

Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.