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

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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,

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

Therefore

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

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 1)

And substituting in equation 1, we get

When or (Equation A)

When or (Equation B)

When or (Equation C)

We can be rewrite equation A as say 3m

Where,

Also equation B can be written as or

Where,

Also equation C can be written as or

Where,

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

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

Solution:

We know that by using Euclid’s Division Algorithm,

Substituting , we get

Where,

When

, where

When

, where

When

Continuing the process till , we get

Where,

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

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