p be a prime number. Show that for any integer a the greatest common divisor of p and a is either p (if pla) or is 1 (if p does not divide a).

Q: (a - b)(a − c)(a − d)(b − c)(b − d)(c - d) -

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Q: Let A is multiples of 3 then show that it is countable infinite

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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: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

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Q: Find if 5, 14, and 22 are pairwise relatively prime.

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Q: If x is divisible by 5 or y is divisible by 5, then xy is divisible by 5.

A: Let x is divisible by 5. Then x=5u for some natural number u.

Q: An integer is square-free if it is a product of distinct primes. Show that for each integer k≥ 1…

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Q: Find the value of the permutation 2Pi

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Q: a. Place the digits 7, 6, 8, and 3 in the boxes to obtain the greatest product. b. Place the digits…

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Q: Use the factorization A = PDP to compute AK, where k represents an arbitrary positive integer. - 7…

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Q: Consider 36, 756, and 1001.

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Q: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: Show that if p is a prime number, a and b are positive integers and p | ab, then p | a or p | b.

A: The objective is to prove that if p is a prime number, a and b are positive integers and p | ab,…

Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

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Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

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Q: a = ? b = ? c = ? d = ? e = ?

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Q: For each pair of integers r and s below, Find the greatest common divisor d. Then, find m, n, such…

A: To determine the greatest common divisor, d and then to find m and n such that d = mr + ns. (a) r=32…

Q: Determine the number of ways a computer can randomly generate one or more such integers from 1…

A: There are 5 prime numbers between 1 to 12 . Which are 2 ,3 , 5, 7 , 11.

Q: Let m and n are relatively prime, positive integers. What is the greatest common divisor of 5m + 7m…

A: Since m and n are relatively prime, we can use the following property: For any two relatively prime…

Q: C. Find the possible values of a so that an + an+1 + an+2+ an+3 + an+4 will be divisible by 11.

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

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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: Determine which positive integers ? have a. Exactly 2 positive divisors b. Exactly 4 positive…

A: A positive integer has exactly 2 positive divisors if and only if it is a prime number.

Q: Show what the divisors of 120 are and what they add up to.

A: Divisors of 120=1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

Q: Show that if 2" – 1 is a prime number, then n is a prime number.

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Q: Find the prime factorization of (2) and of 20!. 10

A: To find the prime factorization of 2010 and of 20!. Prime factorization means we have to represent…

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

A: Given ,  For each positive integer n , let p(n) be the property 5n - 1 is divisible by 4

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…