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

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: If a and b are two distinct prime numbers, then the greatest common divisor of a and b is O ab

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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: (a) If p is a prime number and a € Z such that pla, determine whether pl(a + 1) (b) Show that the…

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Q: Let be the prime number p>5. prove that the expression p^4-10p^2+9 is divisible by 1920

A: First we will expand the given expression p4-10p2+9⇒ p4-10p2+9=p4-9p2-p2+9⇒…

Q: If a is divisible by b, then bq, for some integer q. a = ulgn O ihi O

A: We have to check whether the statement, "If a is divisible by b then a=bq for some integer q" is…

Q: Suppose a, b, and c are integers and suppose a | b and a{c, where a}c denotes that c is not…

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Q: Show that if m and n are integers and a and b are elements from aring, then (m . a)(n . b) = (mn) .…

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Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

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Q: 1. |x+y|≥x+y 2. 3=5, or 7 is a prime number. 3. If x is a prime number, then x is odd.

A: 1) For first statement we can show that by modulus of a number   2) One of the substatement is true…

Q: show

A: Given : m+n and n+p are odd  So, we can write, m+n = 2s+1 and, n+p = 2t+1, where, s & t are…

Q: 5. Let a, b, c, and d be any four integers. Assume that 2 divides a, that 3 divides b, that 5…

A: If x divides y, it can be written that y=xk, where k is a constant. In other words, if y is divided…

Q: How many integers from 1 to 1000: a) Are divisible by 7? b) Are divisible by 7 but not 11? e) Are…

A: a) From 1 to 1000 there are 1000 numbers. Now, 1000/7 = 142.86 So, there are 142 multiples between 1…

Q: For any integer ? show that the pair of integers 7? + 9 and 4? + 5 are relatively prime.

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Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: Replace the values of n by given values

Q: b) If x is an even prime number, then x is divisible by 2. Converse: Inverse: Contrapositive:

A: Given the statement, "If x is an even prime number, then x is divisible by 2". It is of the form,…

Q: If p is a prime number greater than 3, which of the following represents the sum of the positive…

A: we will check all the options to ensure which is correct.

Q: How to solve this?

A: According to the given information, let p be a prime number such that p>2. It is required to…

Q: If p is prime and pl ab then p la and plb Select one: O True O False

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Q: Given: " if n is an integer and 3n + 2 is even, then n is even" can be proven by both direct proof…

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

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Q: which of the below integers is congruent to 6 modulo 12 O none 40 30 34

A: 30

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

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Q: 14. Show that ifr>0, then, for each natural number n, (1+r)" >1+n•r.

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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: integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

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Q: Is the following proposition true or false? Justify your conclusion with a proof or a…

A: Check the following proposition, "For each natural number n, if 3 does not divide (n2+2) then n is…

Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: Find out how many distinct three-digit numbers can be formed using al the digits of 1, 2, and 3. * O…

A: To find  Number of distinct three digit numbers can be formed using all the digits 1,2 and 3.

Q: Suppose a, b, and c are integers and suppose a |b and ałc, where a{c denotes that c is not divisible…

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

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Q: If a is divisible by b, then bq, for some integer q. a

A: See the detailed solution below.

Q: The number of incongruent primitive roots modulo a prime p is: a) Ø(p) b) Фф — 1) с) р—1 O a O b

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

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Q: If an integer n is odd, then n2 + 2n +1 is even. Select one: True O False

A: Given n is odd Now n2 + 2n + 1 = (n+1)2

Q: Assume a,b,p € Z and p is prime. Prove that if plab, then pla or p b.

A: We have to solve given problem:

Q: If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

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Q: If p2q25 is a prime number, prove that p-q² is divisible by 8.

A: Given, p≥q≥5 are prime numbers, Now, p2-q2=p+qp-q. Since, p and q are prime which are grater than…

Q: If p is prime and k > 2, then Ø(Ø(p*)) %3D a) pk-2ø((p – 1)²) b) pk-1ø((p – 1)²) k-20n

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Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: Show that sqrt(3) is an irrational number. (And show and use that if 3 divides n^2, then 3 divides…

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Q: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

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

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Q: Every natural number n > 2 cam be writtenn = PiP2 Pr as the ... product of distinct prime numbers…

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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: If X is a positive integer divisible by 6, and X <60 What is the greatcst possibe vaive of X 7

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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

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Q: For all prime numbers  a, b, and c, (a^2 +b^2) does not equal c^2. prove by contradiction

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.