Prove or disprove: For any integers a,b,c, if a { b and b X c, then a X

Q: If a, b. and c are integers such that a|b and c|b. then ac|b.

A: False

Q: Prove by CASES: Let x be an integer. If 3 does not divide x, then 3 divides x²−1.

A: Given x is an integer. We have to prove that If 3 does not divide x, then 3 divides x²−1.

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

A:

Q: Prove the following statements: iii. Suppose that a, b, c and d are positive integers, prove that…

A:

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).…

A:

Q: If a, b, and c are integers such that a|b and c|b. then ac|b. ylg

A:

Q: Prove or disprove: Let a and b be integers. If a divides b and a divides c, then a divides b-c.

A: Let a, b, and c are integers. Integer a is a factor of b and c.

Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

A:

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Prove by any method(contradiction/contrapositive or cases) Prove that if a and b are positive…

A: Prove that if a and b are positive integers such that a|b and b|a, then a=b Let a and b are positive…

Q: Determine whether the following statements are true or false. Justify answer with a proof or a…

A: Introduction: In Mathematics, the term "integer" was adapted from Latin. Integer means complete or…

Q: Z+ is the set of positive integers. Let A = {1,2,3), B = {n ∈ Z+ | n is even}, C = {n ∈ Z+ | n is…

A: concept Used: C∩D is the set  containing all elements of C that also in the set D C∪D is the set…

Q: Prove or disprove: For all integers a and b, if a+b is even, then a is even and b is even. Provide a…

A: Given statement is false because

Q: (b) Suppose a, b, and c are integers and suppose a | b and a{c, where a{c denotes that c is not…

A: Given, (i) a divides b  (ii) a doesn't divide c  To show : a doesn't divide (b+c) On the contrary,…

Q: Let a, b, and e be integers, where a # 0. Then (i) if alb and alc, then al(b + c); (ii) if alb, then…

A:

Q: (b) If a, b and c are integers such that b| a and b| c, then b | (2a – 3c).

A: We have to proof

Q: For all integers a, b and c, a|bc if and only if a|b and a|c True or False

A:

Q: Prove or Disprove: If x is an integer and 7x³+9 is even, then x is odd.

A: According to the given information, it is required to prove or disprove the given statement.

Q: Prove or disprove the following statements: 1. a) For all nonzero integers a, b, c if a | c then ab…

A: Dear students, you have asked multiple questions, as per our policy only first question will be…

Q: (a) For all integers a, b, and c with a # 0, if a | b, then a | (bc).

A:

Q: let a and b be integers. Show: if a * b is odd, then a is odd and b is odd

A: A number is even, if it is divisible by 2. A number is odd, if it is not divisible by 2. The odd…

Q: a.) Let a and b are positive integers. Prove that (a,b) = 2(a/2, b/2).

A:

Q: For all positive integers a, b, where a < b, if gcd(a, b) = 1, then a b. O True O False

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

Q: Prove or disprove. There are integers a, b, and c such that a divides bc, but a does not divide b…

A: We have to prove or disprove that if integers a, b, and c such that a divides bc,but a does not…

Q: Prove or Disprove: If x and y are integers and x - y is odd, then x is odd or y is odd

A:

Q: Let a and b be integers. Proof  by contrapositive that a + b is an even number if and only if both…

A:

Q: If a, b. and c are integers such that alb and c|b. then ac|b.

A:

Q: Suppose that a, a', b, b' are positive integers satisfying a) ab' a'b = b) gcd(a, b) = gcd(a', b') =…

A: Given that a,a',b,b'are positive integers such that  ab'=a'bg.c.da,b=g.c.da',b'=1 We have to show…

Q: Prove by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

A:

Q: Prove If a,b, and c are integers with a|(b+c) and a|b, then a|c.

A:

Q: Disprove: For all integers a,band c, if a|(bc) then a|b and a|c

A:

Q: Which of the following statements is true: For all integers a, b and e, if (a + b)lc and then a | c…

A:

Q: a. Let, a and b be integers such that 1 ≤ a ≤ b. If b = aq + r, where 0 ≤ r ≤ a, then prove that,…

A: What is Greatest Common Divisor: For two integers, their greatest common divisor is an integer that…

Q: (d))For all integers a, b, and c, if a | b or a c, then a | bc (e) For all integers a, b, and c, if…

A: (d) let for all integers a,b,c if a|b or a|c so a|b or a|c ⇔b = ak1 or  c = ak2  for some integers…

Q: Let n, a, and b be positive integers such that ab = n. If (a, b) = 1, prove that there exist…

A:

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 or disprove: If a and b are odd integers, then 4 | (a – b) or 4 | (a+b). -

A:

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

A: prove is given below ....

Q: prove For all integers a, b, c, and d, if a|c and b|d then ab|cd.

A:

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).

A:

Q: Prove or Disprove the following statement there are no positive integers such that x^2 - y^2 = 10

A: We prove this statement by using contradiction method. We assume two cases and show that we get some…

Q: rove that if x and y are odd positive integers ,then x+y is even but not divisible by 4

A:

Q: Prove or disprove: For all integers a and b, if a +b is odd, then a is even and b is odd.

A:

Q: and

A:

Q: Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

A: We disprove this statement by taking a counter example.

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!