For all odd prime numbers p, ((²¹)!)² = (−1)²+¹ mod p.

Q: 98! =... .(mod101) (а) 60 (b) 50 (c) 51 (d) None of these Select one: • d x

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Q: For integers a and b, prove that (a + b) 3 a3 + b3 (mod 3).

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Solve 57x ≡ -3 (mod 50) for x

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: If you multiply n whole numbers (even or odd) when is the answer odd? Translate into multiplication…

A: If we multiply whole numbers, the pattern is as follows: Even × Even=Even Even × Odd=Even Odd ×…

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: Show that if p = q + 4a (p and q are odd primes), then (p/q) = (a/q).

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Q: Podasip 11. If p is prime, then every c can be expressed as a sum of two squares: x² + y² = c(mod…

A: The given is statement is not accurate. The correct statement in connection with the given statement…

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: p, q, pq | (apq − ap − aq  + a) for all a.

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Q: The remainder when aa is divided by 77 is 22 and the remainder when bb is divided by 77 is 55. (So a…

A: Given:          a≡2 mod(7)  and b≡5 mod(7)

Q: n²+3 4 Prove that if n is an odd integer, then √2-2

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Q: Use (9.11) and (9.12) to simplify (ABTC)T, (С-1мс) -1, (АН) -"(АНА-1)"(НА-1) -1.

A: For (ABTC)T by using (9.11) we get (ABTC)T = CT(BT)TAT As (BT)T = B Hence (ABTC)T = CTBAT

Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

Q: Compute 21200 mod 27.

A: We want to compute 21200 mod 27.

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

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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: For each integer n>1, let A(n) denote the sum of the integers from 1 to n. For example,…

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