Problem 5. Find all primes p such that 1+p: 2P is a perfect square.

Q: 6. Determine the number of pairs (a, b) of positive integers with a ≤ 100 and b ≤ 100 such that 324…

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Q: Use the factorization A = PDP 5 -6 = 12:31 I 2 - 2 AK- = 3 2 10 2 1 02 2.2k-3 6-6.2k 2k-1 2-3.2k 1…

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: (6) In your computations below, do the work "without a calculator," i.c., do your computation in a…

A: One may find here that

Q: The nth row of Pascal's triangle lists the coefficients of (x +y)^.

A: Here draw pascal triangle for some 1 to 4 rows and conclude for nth row

Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: (7) Recall that vp(n) denotes the highest power of the prime p that divides the nonzero integer n.…

A: Given that denotes the highest power of the prime p that divides the nonzero integer n.First we have…

Q: 1 Use the factorization A = PDP' to compute A^, wherek represents an arbitrary integer. а 11(b - а)…

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Q: 4. Prove that the product of any two consecutive integers has the form 3k or 3k+2 for some integer…

A: we are entitled to solve only 1 question at a time so I am providing you the same. Kindly repost the…

Q: G. Recall that for any modulus n the resulting remainder can only be within the range of 0 to n- 1.…

A: A detailed solution is given below.

Q: Suppose n is a natural number greater than 100. Alex and Jerry each compute the gcd(4n, 20n2 + 15)…

A: Given that n is a natural number greater than 100. Alex and Jerry computed the gcd4n,20n2+15 and got…

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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Q: A perfect cube is an integer x such that there exists a an integer y and x = y³. Prove that the sum…

A: Let n,n+1,n+2 are the three consecutive integers. consider n=1 therefore,  13+23+33=1+8+27=36 which…

Q: Problem 3. Find all the positive primesp such that 9p +1 is a perfect cube. Namely, such that there…

A: A prime number is such a number which is divisible by 1 and the very number. Prime numbers play a…

Q: that the only perfect number of the form n = pq where p and q are distinct primes is 28.

A: