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

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

A: False

Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

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

A: To check the given statement true or false lets take look a couple of example as follows : Take a=2…

Q: Prove that |ab|= |a | |b| for any numbers a and b.

A:

Q: The question is Prove or disprove: for all integers a, b, c, d, if a divides both b − 2c and c − d,…

A:

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

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

A:

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

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 and m, if a and m are the lengths of the sides of a right triangle and…

A: Given: a and m are length of sides of right triangle and m+1 is hypotenuse then a is an odd integer…

Q: Prove that if a > 2 and b are integers, then a † b or a { (b+1).

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

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: 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 the following statement: "Let a, b and n be integers. If n doesn't divide ab, then n doesn't…

A: The given statement is -" Let a.b and n are integers. If n doesn't divide ab, then n doesn't divide…

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

A:

Q: Show via proof that for two positive integers a, b where a 2 b, (use combinations e.g. n choose k)…

A: We shall prove for the given statement. Though it has little ambiguities.

Q: Show that if a, b, c, and d are integers with a and c nonzero, such that a|b and c|d, then ac|bd.

A:

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: If a, , and c are integers such that albc, then alb or alc. :Select one True False

A:

Q: Prove the statement: For all integers a, b, and c, if a² + b² =

A: Introduction: Proof by contradiction (also known as indirect proving or the method of reductio ad…

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

A:

Q: Prove that If k, q, w and r are integers, and if c|k and c|q, then c|(kw + qr)

A: We know that  Definition of Divisibility : If a and b are integers such that a≠0, then we say "a…

Q: If all entries of A and A -1 are integers, prove that <let A = 1 or -1. Hint: What is det A times…

A:

Q: Prove by contraposition: if a, b and e are integers and be is not divisible by a, then b is not…

A: Given a,b and c are integers. Statement : P: bc is not divisible by a. Statement: Q: b is not…

Q: Suppose that a, b, and c are integers. Prove that if a2 +b2 =c2 then a or b is even.

A:

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

A:

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: If a, and c are integers such that albc, then alb or alc. :Select one True False

A: Given integers 'a' 'b' and 'c' such that a|bc then a|b or a|c. This statement is false.  Let us…

Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: Show that if a and b are positive integers, then ab = lcm(a, b) . gcd(a, b).

A: Consider two cases: 1. a and b are multiples of n i.e. they have a common factor n So a=nx, b=ny…

Q: Prove that if a and b are integers such that a|b, then either a = b or a = -b

A: Take a = 2 ,b = 4 . We have a divides b but neither a = b , nor a = -b.

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

A:

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

A:

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: Are there numbers a and b such that |a + b| > |a|+|b|?

A:

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

A:

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

A: Suppose a|b and a|c. Then there are integers m and n such thatb = am and c = an.Then b + c = am + an…

Q: Let a, b, and p be positive integers. (a) Prove that if pjab ) (pja or pjb), then p = ab ) (p = a…

A: let a,b,p be positive integers. (a) to prove that if p|ab⇒p|a or p|b then p=ab⇒p=a or p=b as given…

Q: Prove that, if a, b, and c are real numbers such that a <b< c, then |b| < max{Ja|, Jel}.

A: Consider the given: a, b ,c are real numbers  such that a&lt;b&lt;c

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

A:

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

A:

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

A: We have to prove the following statement -  If a,b, and c≠0 are integers, then a∣b if and only if…

Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: 2. Given integers a, b, c, d, verify the follOWilng. (a) If a | b, then a | bc. (b) If a | b and a |…

A: Note: according to our guidelines we can answer only first three subparts Given integers a, b, c, d,…

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.

A:

Q: and

A:

Q: Prove that if p,q, and r are integers, r|p and r|q, then r^2|pq.

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.

Q: (a) For integers a, b, q, r, prove that if a = bq + r then (a, b) = (b, r).

A: Since you have posted a multiple question according to guildlines I will solve first(a)    question…

Q: Show that if a | b and b | a, where a and b are integers, then a = b or a = −b.

A: We show that if a | b and b | a, where a and b are integers, then a = b or a = −b.