If a is divisible by b, then bq, for some integer q. a

Q: Let m be the least common multiple of a and b. Prove that if p is a prime divisor of m, then either…

A:

Q: Let a, b, c, d be non – zero distinct digits. The number of 4 digit numbers 'abcd' such that ab + cd…

A: The following results are to be used are as follows: (i) The sum of two even numbers is even  (ii)…

Q: Show that if a and b are positive integers, then there are divisors c of a and d of b with (c, d) =…

A: image is attached

Q: For any nonzero integers a, b, c, k. One of the following statements is true O A. If alb, then |a| <…

A:

Q: If a is divisible by b, then bq, for some integer q. a = ulgn O ihi O

A: We have to check whether the statement, "If a is divisible by b then a=bq for some integer q" is…

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: Given that a and x are integers, a > 1, a | (11x + 3), and a | (55x+52), find a.

A:

Q: Let a be a positive integer. Then Lcm(2a+1,2a+2)= *

A: The given numbers are 2a+1 and 2a+2. Clearly, the given numbers are two consecutive numbers.

Q: Suppose a and ß are transpositions, then show that if aß= (1), then a = B.

A:

Q: If a, b, and c are integers such that b is a multiple of a³ and c is a multiple of b², then c is a…

A:

Q: a

A:

Q: In real analysis,Prove that if A is countable, then m(A) =0.

A:

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: a. Show that if a, b, c, de Zwith a +b√2 = c+d√2, then a = c and b=d. You may use, without proof,…

A:

Q: For each integer n, let T, = {n, nº}. How many elements are in each of T2, T–3, Tị, and To? Justify…

A: To find:The elements in T2, T-3, T1 and T0.

Q: Apply any method to prove that if a|b and alc then a|(bc – 3c)

A: We will use direct proof method to prove the given statement

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

A:

Q: Prove that if r < s < n, then P(n, s) is divisible by P(n.r).

A: Use the formula of permutation

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

A:

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

A:

Q: Show that ∗ : R × R → R given by         (a, b) → a + 4b2 is a binary operation.

A: Given for all real number a and b

Q: Let a, b and m be integers. Prove that if 2a + 3b ≥ 12m + 1, then a ≥ 3m + 1 or b ≥ 3m + 1.

A: Here I have corrected the statement of the Question and provide a detailed solution of the problem.

Q: If a is any integer, is a(a + 1) even or odd?

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: Let a = and b = . Calculate the following: compьa proj, a

A: From the given problem:   a=&lt;-3,1,3&gt; , b=&lt;4,-2,-1&gt;   As we know that:…

Q: If a is divisible by b, then bq, for some integer q. las

A: If a is divisible by b, then, a = bq for some integer q.  TRUE

Q: If b is a positive integer not divisible by the prime p, then (р-1)b 2b 3b + %3D ...

A: We have to find the required summation value.

Q: Prove that for any integer a , one of the integers a,a+2,a+4 is divisible by 3

A:

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: Consider these four statements: A: For all integers a, b, and c, if a | b and a | c then a|(2b –…

A: The objective is to find the correct statement.

Q: Prove that every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

A: Let P(n) be the statement, "For n∈ℕ there exist a, b&gt;0 such that              n+27=a·5+b·8 Use…

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: If two rows of A are interchanged to produce B, then det(B) = –det(A). \\ Big True fals e

A: We use properties of determinant

Q: If m and k are positive integers, then, then Ø(m^k )= *

A: The objective is to state whether the statement is true or false. a) The statement: The quadratic…

Q: For every positive integer a, find a composite number n such that n|a" – a.

A:

Q: Suppose a is divisible by b. Show that gcd(a, b) = b.

A: Consider the given information. As we have given that a is divisible by b. then there is a number c…

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: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

A:

Q: Prove the given statement: If the product of two integers is not divisible by an integer n, then…

A:

Q: Suppose that [E:Q] = 2. Show that there is an integer d such thatE = Q(√d) where d is not divisible…

A:

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

A:

Q: Prove: If a &gt; b, then a &gt; 1/2 ( a + b ) &gt; b.

A:

Q: prove if a and b are both odd integer, then 16/ (a^4) + (b^4) -2.

A:

Q: If A n CCB n C and AnCcBnC, then show that ASB.

A: Introduction :

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

A:

Q: Let suppose integers are randomly selected. Prove that the difference of two of them is divisible by…

A: Solution

Q: For natural numbers k,m,n, and p, let k be the greatest common divisor of m and n, and let p<k. The…

A: Theorem: The greatest common divisor of m and n is the least positive value of an+bm where a and b…

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.