Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least one of the numbers a, 6, c is divisible by 3.

Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

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Q: Prove that for any integer a, 3 divides

A: To prove that for any integer a, 3 divides (a-3)(a+4)(a+2).

Q: For any integer k, prove that gcd(2k +3, 5k +7) = 1.

A: Use the given information.

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A: Either n is a prime or it is composite; in the former case, there is nothing more to prove. If n is…

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Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Let a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

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Q: Let a,b,c, p be rational numbers such that p is not a perfect cube. If a + b p3 +cp = 0, then prove…

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Q: a

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Q: |Let a, b, and c be integers. Prove that if gcd(a, b) | bc, then a | 1 and gcd(a, c) 1 and a (a) C.…

A: To prove the required divisibility properties under the given conditions on the greatest common…

Q: If a, b, c are integer powers of 2, prove that equation 3 a° + bª = c5

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Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

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Q: Prove that for every positive integer m, there is a positive integer k such that k does not divide m…

A: Given:-                 prove that every positive integer m, there is a positive integer K such that…

Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: Let a and b be natural numbers, where a,b >= 2, such that gcd(a,b)=1 and ab=(x+y)n for some natural…

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r + y-3 is…

A: Solution is in step 2

Q: Prove: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

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Q: Prove that for any integers a and b, if a b and alc, then a|(3b-5c).

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Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

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Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

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Q: For any integer k, prove that gcd( 5k+3, 7k+4 ) = 1

A: Answer

Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

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Q: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

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Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: Prove that if c is a natural number then for all a and b we have : gcd(ac, bc) = c gcd(a, b)

A: We have to prove that if c is a natural number then for all a and b gcd(ac, bc)=c gcd(a, b) .

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

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Q: Show that the operation a*b=1+ab on the set of integers Z, has no identity element

A: Let e be the identity element which exist.  Then, a*e = e*a = a

Q: . Prove that V6 is irrational number. -. If a, b e R, prove that ab <(a² + b°).

A: We have to prove that 6 is irrational number

Q: A perfect cube is an integer x such that there exists a an integer y and x = y³. Prove that the sum…

A: Let n,n+1,n+2 are the three consecutive integers. consider n=1 therefore,  13+23+33=1+8+27=36 which…

Q: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

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Q: Let a, b, c be non-zero integers. Prove that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a,…

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Q: Prove: For all integers a, b, and c, if alb and alc then a (2b+3c)

A: prove is given below ....

Q: (a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers.…

A: According to the guidelines I am answering first question only. a) a, b and c be integers such that…

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

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Q: 2. Show that if a and b are positive integers and alb, then a < b. 3. Show that if a and b are…

A: Basic facts about division will be used in solving this problem.

Q: 6. Prove that for all integers a, b, and c, if a² + b² = c², then at least one of a and b is even.

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Q: and

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Q: (a) Any integer with factorisation 30.76.11c where a, b, c E N, is said to be 9-regular if and only…

A: Please see the below picture for detailed solution.