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Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8?
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Use Euclid's division lemma to show that thw cube pf any positive inte...
Let a and b be two positive integergs such that a is greater than b; then :
a = bq + r;
where q and r are positive integers 0 ≤ r < b.
Taking b = 3, we get
a = 3q + r ; where 0 ≤ r < 3.
⇒ Different values of integer a are 3q, 3q + 1 or 3q + 2.
Cube of 3q = (3q)^3
= 27q^3 = 9(3q^3) = 9m ;
where m is some integer.
Cube of 3q + 1 = (3q + 1)^3
= (3q + 3(3q)^2 � 1 + 3(3q) � 12+ l3
[∵ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + 1]
= 27q^3 + 27q^2 + 9q + 1
= 9(3q^3 + 3q^2 + q) + 1
= 9m + 1; where m is some integer.
Cube of 3q + 2 = (3q + 2)^3
= (3q)^3 + 3(3q)^2 � 2 + 3 � 3q � 22 + 23
= 27q^3 + 54q^2 + 36q + 8
= 9(3q^3 + 6q^2 + 4q) + 8 = 9m + 8; where m is some integer.
∴ Cube of any positive integer is of the form 9m or 9m + 1 or 9m + 8.
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Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8?
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Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8?.
Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Use Euclid's division lemma to show that thw cube pf any positive integer is of the form 9m,9m 1, or 9m 8?.
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