NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 1: Real NumbersExercise 1.4 - Long Answer Questions
Class 10 - Mathematics - CHAPTER 1: Real Numbers
Question. 1
Show that the cube of a positive integer of the form \(6q + r\), where \(q\) is an integer and \(r = 0, 1, 2, 3, 4, 5\), is also of the form \(6m + r\).
Answer:
Yes. The cube of such an integer is again of the form \(6m + r\).
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Question. 2
Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Answer:
Exactly one out of n, n + 2 and n + 4 is divisible by 3.
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Question. 3
Prove that one of any three consecutive positive integers must be divisible by 3.
Answer:
Yes, in every set of three consecutive integers, one of them is divisible by 3.
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Question. 4
For any positive integer \(n\), prove that \(n^3 - n\) is divisible by 6.
Answer:
Yes. For every positive integer \(n\), the number \(n^3 - n\) is divisible by \(6\).
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Question. 5
Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.
Answer:
Exactly one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.