Which of the following is true? There are infinitely many primes of the form n2- 9. If lcm(a, b) = b then gcd(a, b) = 1 %3D If p is prime and ab| p then a = 1 or b = 1 If alc and blc, then ab|c None of these

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: i +1 If z = E C, find simple expressions for i and /2 in terms of z. Then prove: If p is an odd…

A: Given, Z=i+12∈ℂ We can write the above equation as, Z2=i+122=i2+2i+12=-1+1+2i2=iSo, i=Z2

Q: Show that for every integer n, either 3 | n or n² = 1 (mod 3).

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Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: 7. Now using 4, 5, and 6, we have N < VN . 21(N). Get II(N) by itself on the right-hand side of this…

A: According to 4: N≤S(N).F(N) According to 5: S(N)≤N for every N≥1 According to 6: F(N)≤2∏(N) Solution…

Q: Icm(a,b) = ab if and only if a and b are relatively prime.

A: We know the relation between product of the two numbers their L.C.M and G.C.D Let a and b two…

Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

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Q: Decide which of the following are valid proofs of the following statement: Let n be any integer.…

A: Let n be any integer. Then n3-n is divisible by 3.

Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

A:

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Time left 1:0 If p and q are twin primes, then which of the following is true. i. p + q is divisible…

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Q: If n is an odd integer, then the floor of n/2 = (n-1)/2.

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Q: Select the following statements that are true. In Z6.2¹0 = 2 261000 = 1 (mod 3) gcd(a+2, a) = 2 for…

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Q: Sophie Germain (1776-1831) studied at the École Polytechnique in Paris in a day when female students…

A: Using the given definition, we can determine whether given p is a Sophie Germain prime or not.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.