Show that if a,b, and c≠0 are integers, then a∣b if and only if ac∣bc.

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: Give examples of fields containing 4, 8, 9, &27 elements.

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Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

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Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: Prove that 1+1=2

A: To prove: 1 + 1 = 2

Q: at any positive odd integer is of the

A: Here we have to show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,…

Q: let a,b,c,d,e and f be different positive integers so that a² - b² = c² - d² = e² - f² = 63. Value a…

A: Given:- a,b,c,d,e and f are different positive integers.  also a2-b2=c2-d2=e2-f2=63. To find :- the…

Q: Assume a, b ER. a+b, a² + b², a³ + b³, and a + b for all integers. Prove that am + bm is an integer…

A: Let us assume that a,b∈ℝ where ℝ is the set of real numbers. It  follows that a,b∈ℤ where ℤ is the…

Q: Prove that 1+6x-x is always a positive real number if x =3+Ai for some real 33 number 4

A: The given number is  1 + 6x - x2

Q: Let a, b, c, m, and n, be integers. Write in complete sentences to prove that if a|b and a|c, then…

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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 any prime number Divide ‘p’ by 6, we get P=6q+r       - (1);     r=1,2,3,4,5 And q be any…

Q: Let а, b and c be integers. Use the method of proof by contradiction to prove that if a|b and a f c,…

A: We have to solve given problem:

Q: Let a, b, and c be integers with a 0. Prove that if a | b and a | (b + c), then a | c.

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Q: If+b+=++, then show that for 1 a" +b" +c"

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Q: Prove that if a, 6, and c are positive real numbers with ab = c, then a < yc or 6 < Vc.

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Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

Q: Prove: Let a,b,x and y be integers. If a|x and b|y then (ab)|(2xy).

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.