NCERT Class 10 Solutions: Real Numbers (Chapter 1) Exercise 1.1 – Part 2

Q-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?

Solution:

For the above problem , the maximum number of columns would be the HCF of 616 and 32

We can find the HCF of 616 and 32 by using Euclid Division algorithm.

Therefore,

Equation

Since remainder Equation , we apply the division lemma to 32 and 8 to obtain

Equation

Therefore Equation

Therefore, they can march in 8 columns each.

Q-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 Equation . Now square each of these and show that they can be rewritten in the form Equation .]

Solution:

Figure explaining two cases of euclids divsion

Euclids Divsion Cases

Figure explaining two cases of euclids divsion

According to Euclid algorithm

We have Equation (Equation 1)

And substituting Equation in equation 1, we get

Equation Equation

Equation

When Equation or Equation (Equation A)

When Equation or Equation (Equation B)

When Equation or Equation (Equation C)

We can be rewrite equation A as Equation say 3m

Where, Equation

Also equation B can be written as Equation or Equation

Where, Equation

Also equation C can be written as Equation or Equation

Where, Equation

Hence, the square of any positive integer is either of the form Equation for some integer m.

Q-5 Use Euclid’s division lemma to show that the cube of any positive integer is of the form Equation .

Solution:

We know that by using Euclid’s Division Algorithm, Equation

Substituting Equation , we get

Equation Where, Equation

Equation

When Equation

Equation , where Equation

When Equation

Equation

Equation , where Equation

When Equation

Equation

Equation

Continuing the process till Equation , we get

Equation

Equation

Equation

Where, Equation

Hence, it is proved that any positive integer is either of the form Equation

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