Please help me thanks! Follow the directions in this section for writing proofs of universal statement. But for this problem do not do any division...work with showing n-1 is not prime. 1. If an integer greater than 4 is a perfect square l, then immediately preceding integer is not prime.

Please help me thanks! Follow the directions in this section for writing proofs of universal statement. But for this problem do not do any division...work with showing n-1 is not prime. 1. If an integer greater than 4 is a perfect square l, then immediately preceding integer is not prime.

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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Prove or disprove each of the following statements. To disprove just give a counter example. Believe the statement is true but not sure how to prove it 2 can be represented as 4 irrational numbers by sqrt(2) * 1/sqrt(2) * 2sqrt(2) * 1/sqrt(2)
Give a direct proof: Ifx is even, then x³ + x + 1 is odd. (Use the definitions of even and odd, and facts about arithmetic of integers. Do NOT use more general statements like "even times even is even". Try to not leave out any steps or reasons!)
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Please prove this question the other way around... 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is even if and only if 4 divides n². (don't prove what is written above, prove it the other way around please)
Another Field We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from 13 2 the set Q). For example, +¹3√2, 7.2-6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √√3 nor are rational numbers). A number such 3+5√2 would be in the set Q(√2) since 3+√2 can be re-written in the form 7 a+b√2 using the following steps: 7 7 3-5√2 21-35√2 21-35√2 21 - 35√2 21 35 == + √2 3+5√2 3+5√2 3-5√2 9 - 25(2) -41 41 41 = * Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and * is the ordinary multiplication operation. So, you need to show that: a. (Q(√2), +, *) is a commutative ring, and b. (Q(√2) - {0}, *) is a group.
3 Show that 2 is irrational.