Theorem 4.12. Gauss: Let n #0 (mod p). Consider the remainders mod p of the following (p-1)/2 many integers: p-1 n. 2n. 2 Let m be the number of these remainders which exceed p/2. Then (=) = = (-1)m. -n.

Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: a. 3.2F> Fn+1 for n23 [Solutions and final ans

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Q: Let m be the number of these remainders which exceed p/2. Then ( #) = (-1)m.

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Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: (4 – 2æ)".

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Q: Let Find formulas for the entries of M", where n is a positive integer. Hint: a formula such as 5…

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Q: Q3. By using mathematical induction, show that 52n – 1 is divisible by 24, for all positive |…

A: We need to show that 52n-1 is divisible by 24. We can solve this using induction method.

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Q: 6. For an arbitrary integer a, verify the following: (a) 2 a(a + 1), and 3 a(a + 1)(a + 2). 4(b)…

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Q: 1a.) 0x**3  + 9x**3  1b.) 4X**3  - 9x**5  1c.) (3x   + 1x**3) + 2(9x**3 +(-3x)) 1d.)…

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Q: x4. Prove (1-1) (1-1)… (1-급) n+1 for all n 2 2. 2n 22 32 X5 From the homework and examples vou have…

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Q: 6.The least positive residue or remainder of 12^20 (mod 25)  a.8 b.6 c.4 d.2 9.Which pairs of…

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Q: Prove the following for Integers x and y, (17 | 10x + 8y) = (17 | 3x + 16y)

A: 17 | 10x + 8y where x and y are integers => 10x + 8y = 17m for some integer m