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If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4 – b4) always divisible?
- a)3
- b)6
- c)8
- d)12
Correct answer is option 'C'. Can you explain this answer?
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If ‘a’ and ‘b’ are two odd positive integers, ...
We know that, Summation and Subtraction between two odd integers gives an even integer.
Given, a and b are two odd positive integers.
∴ a + b = 2k1 ……(1)
And a – b = 2k2 ........(2)
Where, k1 and k2are two integers.
(1) + (2) ⇒ 2a = 2(k1 + k2) ⇒ a = k1 + k2
(1) - (2) ⇒ 2b = 2(k1 - k2) ⇒ b = k1 - k2
∴ a2 + b2
= (k1 + k2)2 + (k1 - k2)2
= k12 + k22 + 2k1k2 + k12 + k22 - 2k1k2
= 2(k12 + k22) ………..(3)
Given expression is,
a4 – b4
= (a2 – b2)(a2 + b2)
= (a + b) (a - b)(a2 + b2)
Putting values from (1), (2) and (3)
= 2k1 × 2k2 × 2(k12 + k22)
= 8× k1 × k2 × (k12 + k22)
∴ a4 – b4 = 8× k1 × k2 × (k12 + k22) is always divisible by 8.
Given, a and b are two odd positive integers.
∴ a + b = 2k1 ……(1)
And a – b = 2k2 ........(2)
Where, k1 and k2are two integers.
(1) + (2) ⇒ 2a = 2(k1 + k2) ⇒ a = k1 + k2
(1) - (2) ⇒ 2b = 2(k1 - k2) ⇒ b = k1 - k2
∴ a2 + b2
= (k1 + k2)2 + (k1 - k2)2
= k12 + k22 + 2k1k2 + k12 + k22 - 2k1k2
= 2(k12 + k22) ………..(3)
Given expression is,
a4 – b4
= (a2 – b2)(a2 + b2)
= (a + b) (a - b)(a2 + b2)
Putting values from (1), (2) and (3)
= 2k1 × 2k2 × 2(k12 + k22)
= 8× k1 × k2 × (k12 + k22)
∴ a4 – b4 = 8× k1 × k2 × (k12 + k22) is always divisible by 8.
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If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer?
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If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer?.
If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4– b4) always divisible?a)3b)6c)8d)12Correct answer is option 'C'. Can you explain this answer?.
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