Reference no: EM13493585
1. Find the unique factorisation of 1176.
2. Calculate the following:
(a) 228 div 119
(b) -10101 mod 333
(c) 230 x 91 mod 11
3. Prove that every number from the set {2, 4, 6, 8, ... , 26} can be written as the sum of at most 3 square numbers (A square number is a perfect square, e.g. 1 = 12,4 =22, 9 = 32, etc.)
4. Prove: ∀ k ∈ Z, [3|(k - 2) -> + 3|(k2 - 1)].
5. Prove the following statement:
∀ a, b, c ∈ Z, if a|b and a|c, then a2|(5b2 + 7c3).
6. Consider the following statement:
∀ a, b ∈ R, [(ab > 1) -> ((a > 1) V (b > 1))]
(a) Write the negation of the statement.
(b) Determine which of the original statement or the negation is true. Justify your answer.
7. Use a direct proof to show that
1+ 2+3+...+n = n(n+1)/2
(You might get some inspiration from the story of little Gauss who was given a similar problem to do in class. Gauss was asked to add the first hundred positive integers. He took no time at all to work it out! How did he do it? (clue in cartoon!) Apply a similar idea to your proof. Later in the course, we will use another method to prove the result).
8. Prove the following:
(a) For any integer a, a ≡ a (mod m), where m is a positive integer.
(b) For all integers a, b, c, if a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m), where m is a positive integer.
(c) For all integers a and b, if a ≡ 2 (mod 4), and b is odd, then ab ≡ 2 (mod 4).
9. Are the following statements true or false? Prove the statement if it is true and give a counterexample if it is false.
(a) [x - 2] = [x] - 2.
(b) For any odd integer n, [n2/4 + 1 ] = n2 + 3 /4.
10.
(a) Prove the Giftbox Theorem using the method of contradiction:
Suppose there are (n+ 1) gifts and n gift boxes. Then some gift box must contain at least 2 gifts.
(b) Use the Giftbox Theorem to prove the following:
Suppose you have a line segment which is 1 metre long. You throw darts at the line segment and being a good shot, your dart lands on the line segment each time. You throw 11 darts in total. Show that there is a pair of darts with distance not greater than 10 cm.
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What is the translational speed of the frozen juice
: Starting from rest, a basketball rolls from the top of a hill to the bottom, reaching a translational speed of 2.32 m/s. Ignore frictional losses. What is the translational speed of the frozen juice can when it reaches the bottom
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Explain the geometric and electronic structures
: The complexes [Mn(H2O)6]2+, [Fe(H2O)6]3+, [MnCl4]2-, and [FeCl4]- all have magnetic moments of nearly 5.92 BM. What does this tell you about the geometric and electronic structures of these complexes
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Explain square planar and which is tetrahedral
: Which of these is square planar and which is tetrahedral? State reasons for your choice. b) Rationalize the colors of these complexes in view of your choice.
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Explain what will be the vapor pressure if the temperature
: Carbon disulfide normally boils at 46.5 degrees celsius. when the temperature is lowered to 28.0 degrees celcius, its vapor becomes 400 mmHG. what will be the vapor pressure if the temperature is lowered some more to 18 degrees celcius
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Prove the giftbox theorem using the method of contradiction
: Prove the Giftbox Theorem using the method of contradiction - Determine which of the original statement or the negation is true.
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Find the maximum height of the rocket
: A two-stage rocket is launched at an initial angle of 83 degrees from the horizontal, and for 15.4 seconds,its engine provides a uniform acceleration in its lauch direction of 25m/s^2. Find the maximum height of the rocket
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Explain what the static ground state structure must be
: Based on the composition and the proton nmr observations, show by the appropriate crystal field splitting diagrams what the static ground state structure must be.
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Explain ions can form complexes with the thiosulfate ion
: We sometimes use complex formation to make relatively insoluble salts more soluble. For instance, AgCl(s) has Ksp=1.6x10^-10. Silver ions can form complexes with the thiosulfate ion
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Calculate the torque created by the force
: A force, F(vector) =(2xˆ+3yˆ) N, is applied to an object at a point whose position vector with respect to the pivot point is r(vector) =(4xˆ+4yˆ+4zˆ) m
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