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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.

Sachin Khanna by Sachin Khanna
May 31, 2020
in NCERT Question Solutions
0

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

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?

Question: 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.

Class 10th
Subject Maths
Chapter Real Numbers
Exercise 1.1
Previous Question 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?
Next Question 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:

Let x be any positive integer and y = 3.

By Euclid’s division algorithm, then,

x = 3q + r for some integer q≥0 and r = 0, 1, 2, as r ≥ 0 and r < 3. Therefore, x = 3q, 3q+1 and 3q+2 Now as per the question given, by squaring both the sides, we get, 2 = (3) 2= 9 2 = 3 × 3 2 Let 3 2 = m Therefore, 2 = 3m ……………………..(1) 2 = (3 + 1) 2 = (3) 2+1 2+2 × 3q × 1 = 9 2 + 1 + 6q = 3(3 2+2q) + 1 Substitute, 3 2+2q = m, to get, 2 = 3m + 1 ……………………………. (2) NCERT Solution for Class 10 Maths Chapter 1 Real Numbers 2 = (3 + 2) 2 = (3) 2+2 2+2 × 3q × 2 = 9 2 + 4 + 12q = 3 (3 2 + 4q + 1) + 1 Again, substitute, 3 2+4q+1 = m, to get, 2 = 3m + 1…………………………… (3) Hence, from equation 1, 2 and 3, we can say that, the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

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

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