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

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: Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

A: Let p be a prime number other than 2 and 3. If p is divided by 6, then by the quotient-remainder…

Q: Every natural number n > 2 cam be written n = P1P2 Pr as the ... product of distinct prime numbers…

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

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Q: Find integers ? and ? such that 3220? + 6006? = gcd(3220,6006).

A: Remark: The greatest common divisor (gcd) of two integers is the greatest integer that divides both…

Q: use prime factorizations to find the gcf of each set of numbers  42 and 70? 27, 45, and 54?

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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: All prime numbers are odd.

A: We have state whether given statement is True or False. We know that  A prime number is a number…

Q: The integers 77 and 105 are relatively prime. صواب İhi

A: By finding the factors of given numbers you can find the solution :

Q: Find all four-digit numbers abba with the property aa . 10b= abba if a is not equal to b.

A: We know if abcd is a four - digit number then, abcd = 1000a+100b+10c+d

Q: If n is a natural number, the number 34n+4 – 43n+3 is divisible by: -

A: Given : To find : A positive number which can divide above expression

Q: If the product of two integers is 2'3*5?7!" and their great- est common divisor is 2°345, what is…

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

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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: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

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Q: If z ≥ 2 is an integer, show that the following numbers are composite. n! + 2, n! + 3, n! + 4, ... ,…

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Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

Q: For any integer a, show that a²-a+7 ends in one of the digits 3, 7, or 9.

A: For any integer a , we have to show that a2 - a + 7 ends in one of the digits 3 , 7 , or 9 . Note…

Q: show that if x is an integer then x^2+x-41= 0 produce prime numbers

A: This question is taken from the quadratic equation in which we have to find the roots of the given…

Q: Every odd prime number is greater than 2.

A: Given: "Every odd prime number is greater than 2." To find: Statement in the form "If p, then q"…

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: If b is a positive integer not divisible by the prime p, then (р-1)b 2b 3b + %3D ...

A: We have to find the required summation value.

Q: Find all base 5 digits X such that (124X213)5 is divisible by (2)5 3. а. b. (4)5

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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: Use forward reasoning to show that if x is a nonzero real number, then x2 + > 2.

A: The detailed solution is as follows below:

Q: А. If p 2q 5 is a prime number, prove that p2-q? is divisible by 3.

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

A: Given:

Q: Let A, BeM,(C). Prove that for every integer k21, tr(AB) =tr(BA) Does tr(AB) =tr(A*B* )? Explain…

A: We will do both these subparts of this question for you. Solving this question requires basic…

Q: If n is a perfect number, prove Σ d|n 1 2.

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

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Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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

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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: How many negative divisors has the integer 90?

A: The divisors of 90 are all the numbers that we can evenly divide into 90.

Q: Find all base 5 digits X such that (124X213)5 is divisible by а. (2)5 b. (4)5 3.

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Q: Suppose p and q are distinct prime numbers. Find the divide x. 1. number of positive integers r < pq…

A: Given that p and q are distinct prime numbers. To find: number positive integer x ≤pq such that…

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

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Q: Give a counterexample to the statement. The product of 2 primes is always odd.

A: Product of two primes.

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

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: 2 ) + ("**) = n²- .2 2

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Q: 4. Find them if: (a) lcm of two consecutive positive integers is 812; (b) lcm of twin primes is 899.

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