5. Let n be a positive integer and p1,P2,P3, . „Pn be n distinct primes. If m is a positive integer and the prime factorization of m is pj'pp3 of m3 and m4? ... · P", what is the prime factorization ...

Q: 7. The number 2022 has three distinct prime factors, a, b, and c. The number 2022! can be factored…

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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…

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Q: If a and b are positive integers, which of the following can never be the gcd(56a, 63b)? (а) 56а (b)…

A: given a and b are positive integers,  and to find gcd(56a, 63b) gcd of (56, 63) is 7  so gcd(56a,…

Q: You are competing against all other students in your class. Each of you will choose an umber…

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Q: If fN is the set of the first 100 positive integers, A is the set of odd numbers between 50 and 90,…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

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: Let p, q, and r be prime natural numbers with p # q and n= p2q. Then, all the subfields of GF(r")…

A: Answer: Option (C) Given that, p,qand r are prime natural numbers and n is positive integer. Let F…

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: How many mombers between 50 and 200 can be expressed as 4k-1, where k is an integer?

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Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: 1/q + 1/r + 1/p =1/m p, q, r are pr

A: Given: An expression 1q+1r+1p=1m, where p, q, r be the primes and m be the natural numbers. To find:…

Q: Show that 2 + V5 is imational number

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Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: hree young brother-sister pairs from different families need to take a trip in a van. These six…

A: Given: n=6 Three young brother-sister pairs  Consider a is the first pair, b is second and c is the…

Q: Show that 3 + V5 is irrational number

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Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: Let ß = (1,3,5,7,9,8,6)(2,4,10). What is the smallest positive integern for which ßn = ß-5?

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Q: 2. Factor 10+1 into a product of primes for k = 1,2,3,4,5.

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Q: Let S be the set of numbers of the form { ((-1)^(n + 1)) (n - 1)/(n + 1) } where n is an element of…

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Q: 6. For an arbitrary integer a, verify the following: (a) 2 a(a + 1), and 3 a(a + 1)(a + 2). 4(b)…

A: a is an arbitrary integer.

Q: If p, q, r are prime numbers such that pqr = 19(p + q + r) then the value of 2(p + q + r) is

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Q: Let m = p^3 qr^3 and n = p^2 r^5, where p,q,r are distinct primes. Find each of the following. a.…

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Q: 19. Let N be the positive integer 7777...777, a 313-digit number where cach digit is a 7. Let f(r)…

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Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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Q: Prove that 2 divides a if and only if 2 divides ao. Prove that 4 divides a if and only if 4 divides…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub- parts for…

Q: 7a) Prove that if x is a rational number such that 2x-3 and 5x-4 are integers then x must be an…

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Q: Prove that if p, q, and r are three different prime numbers, then p² +q² # r².

A: WE  prove this statment  by contradiction