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. For any natural number n, prove that (2n) is even. 2. Prove that the product of any five consecutive natural numbers is divisible by 120. 3. Prove that if n is an odd integer, then 8 | (n² - 1). 4. Let a, b e Z with a b and let n € N. Using the binomial theorem, prove that (a- - b) | (a"-b"). 5. Let a, b E Z. Prove that if 4 | (a² +62), then a and b cannot both be odd. 6. A prime triplet is a sequence of three consecutive odd natural numbers that are all prime. For example, sequence (3,5,7) is a prime triplet. Prove that apart from the aforementioned example, there are no on prime triplets. 1. F5 F6 F7 8 I' F8 8 0.

. For any natural number n, prove that (2n) is even. 2. Prove that the product of any five consecutive natural numbers is divisible by 120. 3. Prove that if n is an odd integer, then 8 | (n² - 1). 4. Let a, b e Z with a b and let n € N. Using the binomial theorem, prove that (a- - b) | (a"-b"). 5. Let a, b E Z. Prove that if 4 | (a² +62), then a and b cannot both be odd. 6. A prime triplet is a sequence of three consecutive odd natural numbers that are all prime. For example, sequence (3,5,7) is a prime triplet. Prove that apart from the aforementioned example, there are no on prime triplets. 1. F5 F6 F7 8 I' F8 8 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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. For any natural number n, prove that (2n) is even. 2. Prove that the product of any five consecutive natural numbers is divisible by 120. 3. Prove that if n is an odd integer, then 8 | (n² - 1). 4. Let a, b e Z with a b and let n € N. Using the binomial theorem, prove that (a- - b) | (a"-b"). 5. Let a, b E Z. Prove that if 4 | (a² +62), then a and b cannot both be odd. 6. A prime triplet is a sequence of three consecutive odd natural numbers that are all prime. For example, sequence (3,5,7) is a prime triplet. Prove that apart from the aforementioned example, there are no on prime triplets. 1. F5 F6 F7 8 I' F8 8 0.
Transcribed Image Text:. For any natural number n, prove that (2n) is even. 2. Prove that the product of any five consecutive natural numbers is divisible by 120. 3. Prove that if n is an odd integer, then 8 | (n² - 1). 4. Let a, b e Z with a b and let n € N. Using the binomial theorem, prove that (a- - b) | (a"-b"). 5. Let a, b E Z. Prove that if 4 | (a² +62), then a and b cannot both be odd. 6. A prime triplet is a sequence of three consecutive odd natural numbers that are all prime. For example, sequence (3,5,7) is a prime triplet. Prove that apart from the aforementioned example, there are no on prime triplets. 1. F5 F6 F7 8 I' F8 8 0.
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