m and n be positive integers so that m n. Show for any integer a that the congruence

Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: b.) Show that if a and b are positive integers where a is even and b is odd, then (a,b) = (a/2, b).…

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Q: Prove: there exist no integer a and b for which 21a + 30b = 1.

A: We can prove this by contradiction. Take the negation of the statement and check if it is true, if…

Q: Find all integers z, positive or negatives, such that z2 + z is prime.

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Q: Find the minimum amount of fencing required to surround a rectangu- lar area of 100 square feet.…

A: To solve these problems we use the definition of maxima/minima and absolute maxima/minima which i…

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: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Which of the following is a binary operation on the set of positive real numbers? a. a*b = √a + √a+…

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Q: a

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Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n > 0.

A: Let a be a non zero real number , where m and n are integers…

Q: x³-7x² +10x x²+x-6 4. lim: x-2 5. lim sin лw w4 w²2²-16

A: The objective of the question is determine the indeterminate form of given limits.

Q: Let a and b be positive integers. Suppose Q is defined recursively as follows: if a < b Q(a, b) =…

A: Given Qa,b=0   if a&lt;bQa-b,b+1  if b≤a.

Q: Find the smallest positive integer x such that 11x + 567y = 5 for some integer y. Enter x below.

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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: Use proof by contradiction to show that no integers  y and z exist for which 24y + 12z = 1.

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Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For…

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Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

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Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: Let a and b be two positive real numbers. Let p > 1 and q> 1 be such that + = 1 (q is called the…

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Q: Find ntegers G and t such that gcd (963,652)3967+t-652,

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Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Find all zero divisors in each of the following Z„. f. Z20 d. Z12 e. Z18

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Q: Solve for x. √/2x² = x¹/3 O X = ○== 1 X x 2 Ox ○ x = U, x = = 0 1 29 X = x = 0, x = a = ㅎ H|2 를

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Q: Prove that the number V7 lies between 1 and 2.

A: