Let p2 q be primes greater or equal to 5. Show disprove that 24 divides p² – q². or

Q: A. If p + 2 is a prime number, prove that p? + 3 is divisible by 4.

A: Let p≠2 be a prime number.

Q: 19. What is the value of (pk) when p is prime and k is a positive integer?

A: We know that the value of the Euler ϕ-function at the positive integer n is defined to be the number…

Q: Let p > 5 be the prime number. Prove that the expression p – 10p +9 is divisible by 1920.

A: We have to prove that the expression p4-10p2+9 is divisible by 1920.

Q: Use permutations to solve. From a cooler with 6 different kinds of soda, 4 are selected for 4…

A: nPr = n!/(n-r)!

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

A:

Q: p + 2 is a prime number, prove that p2+3 is divisible by 4. (2.5

A: Prime and divisibility

Q: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: How many integers 2 ≤ n ≤ 65 have no prime divisor less than or equal to ¹/³?

A:

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: If n is prime, then 2" – 1 is prime.

A:

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

A:

Q: Which positive integer less than 21 are relatively prime to 21? 18 O 19 O 21 O 24 O

A: Which positive interger is less than 21 is relatively prime to 21?

Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

A:

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

A:

Q: Characterize the primes p mod 20 such ial

A: We have to characterize p mod 20 such that -5p=1.

Q: If a, b, c, d are distinct prime numbers with an as smallest prime then a * b * c dis None of the…

A: Given a,b,c,d are distinct prime numbers with a as smallest prime. Then, a*b*c*d=_________

Q: Show that there is a prime number p such that p+4 and p+6 are also prime numbers.

A: Prime number is a number p having only two divisors p and 1 only.

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

A:

Q: If p>=q>=5 and p and are both primes, prove that 24| p^2-q^2.

A:

Q: 24. Show that if p is a prime and k is an integer such that Isksp-1.then p divides ().

A: See the handwritten solution

Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

A:

Q: If a, b, c, d are distinct prime numbers with an as smallest prime then a *b * c dis None of the…

A: Answer

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: Prove or disprove: let n be an integer. if 3 divides n^2, then 3 divides n.

A: Let us consider some cases for proving that n be an integer. if 3 divides n^2, then 3 divides n.For…

Q: describe (a,p)

A: Given, if p is a prime, and a is a positive integer.

Q: Show that there are no prime triples other than 3, 5, 7.

A:

Q: Find all prime numbers pand of for which 2p²q + 45 pp ² = y² 2 де

A: Given:

Q: Suppose that 56 % of students play football, 53 % of students play cricket, and 12 % of students…

A: Let F=students play footballC=students play cricketGiven,P(F)=0.56P(C)=0.53P(F¯∩C¯)=0.12

Q: For all odd primes p, show that p 2(p-3)!+1.

A:

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

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: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 +q2 + r2.

A:

Q: Prove that if p> 3 is a prime number, then 24 | (p2 - 1).

A:

Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

A:

Q: The number of primes of the form n³ – 1 is 1) 1 2) 2 3) Infinite

A: To choose correct option

Q: Is 7, 8, 9, 11 are pairwise relatively prime. صواب ihi

A: Two nos. a and b are said to be relatively prime iff gcd(a,b)=1

Q: Show that the sum of twin primes p and p + 2 is divisible by 12, provided that p> 3.

A: Let p, p + 2 be twin primes. Then p *(p+1) + 1 = p^2 + 2p + 1 = (p+1)^2 ,  a perfect square.

Q: yields a true statement when n is replaced by any positive integer.

A: The above statement is true. Step-2 i explained in detailed explanation

Q: if p> q > 5 and p, q are both primes, prove that 24 | p² – q

A:

Q: Find all numbers p such that all six numbers p, p+2, p+6, p+8, p+12 and p+14 are primes.

A: For the given statement

Q: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

A:

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2 + r2.

A: We have to prove that 3 divides p2+q2+r2, where p,q and r primes other than 3.

Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

A:

Q: Let pand g n=p8ge, be distinct primes. het show that v (^)n.

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.