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

Solution:

Suppose that there is a positive integer ‘a’. By Euclid’s division lemma, we know that for positive integers a and b, there exist unique integers q and r, such that a = bq + r, 0 ≤ r < b

Let b = 3, then 0 ≤ r < 3, that is, r = 0 or 1 or 2 but it can’t be 3 because r is smaller than 3.

So, the possible values for a are 3q or 3q + 1 or 3q + 2.

Now, find the cube of all the possible values of a.

If q is any positive integer, then its cube (let’s call it “m”) will also be a positive integer.

Now, observe carefully that the cube of all the positive integers is either of form 9m or 9m + 1 or 9m + 1 for some integer m.

Let 'a' be any positive integer and q = 3. Then, a = 3q + r for some integer q ≥ 0 and 0 ≤ r < 3

Therefore, a = 3q or 3q + 1 or 3q + 2

Case I: When a = 3q

(a)³ = (3q)³ = 27q³

= 9 (3q³)

= 9m, where m is an integer such that m = 3q³

Case II: When a = 3q + 1

(a)³ = (3q + 1)³

(a)³ = 27q³ + 27q² + 9q + 1

(a)³ = 9(3q³ + 3q² + q ) + 1

(a)³ = 9m + 1, where m is an integer such that m = 3q³ + 3q² + q

Case III: When a = 3q + 2

(a)³ = (3q + 2)³

(a)³ = 27q³ + 54q² + 36q + 8

(a)³ = 9(3q³ + 6q² + 4q) + 8

(a)³ = 9m + 8, where m is an integer such that m = 3q³ + 6q² + 4q

Thus, we can see that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

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Video Solution:

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 Solutions Class 10 Maths - Chapter 1 Exercise 1.1 Question 5

Summary:

Using Euclid's division lemma, it can be proved that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.

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