Questions: a. Are there numbers whose squares are smaller than the numbers themselves? b. A prime number is an integer greater than 1 and whose only positive divisors are 1 and itself.

Q: Prove the square of any integer has the form 4k or 4k+1 for some integer k

A: Consider the set of all integers , To prove that the square of any integer is of the form 4k or 4k+1…

Q: How many solutions are there to the equation X1 + x2 + x3 +x4 + x5 = 21 %3D where r;, i =1, 2, 3, 4,…

A: Depending upon the type of equation, the number of solution will vary. If a system contains linear…

Q: Take any arbitrary integer n and then square it, which one is true? n2 cannot be in the form 7k + 5.…

A: Solution is given as follows :

Q: Construct a two-column proof to prove the following by using factoring: Given: a³ - 11a²-9a = -99…

A:

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

A:

Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

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 n is odd, then 2 is not a square in GF(5n). In other words, prove that there is no…

A:

Q: If a, b, c are integer powers of 2, prove that equation 3 a° + bª = c5

A:

Q: Show that if m is an integer then 3 divides m3−m.

A:

Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

A:

Q: Problem 1: Let n and k be positive integers. Show that the multinomial coefficient („ ) (where there…

A: The multinomial coefficient can be resolved as follows: nk1, k2, k3, ⋯ , kr=n!k1!k2!k3!⋯kr!  , where…

Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

Q: Prove that if n is odd, then 2 is not a square in GF(5n).

A: Given n is odd number We have to show that 2 is not a square in GF5n. We know that the field GF5 is…

Q: R.2. Let S represent the set of positive integers that are perfect squares, and let C represent the…

A:

Q: Prove that for all real numbers a, b, c, d, the quantities a – b², b – c², c – ď² , d – a² cannot…

A: We will try to prove the given statement by contradiction.   Let us assume that all of a-b2, b-c2,…

Q: There is a positive integer whose square is equal to itself. real а. There is a number m such that…

A: To rewrite the statement in a different way: Firstly, rewrite the statement in the given statement…

Q: Suppose that a is a positive real number not equal to 1, and that r and s are positive real numbers.…

A: To prove that; rlogαs=slogαr  We have to show that the left hand side of the equation equal to the…

Q: If k is a positive integer, prove the k root of 6 is irrational

A: Given:Given that k is apositive integer.To find:The kth root of 6 is Irrational.

Q: Prove the following using a direct proof: The sum of the squares of 4 consecutive integers is an…

A: To prove that the sum of the square of 4 consecutive integers is an even integer.

Q: Prove that if n is odd, then 2 is not a square in GF(5n) In other words, prove that there is no…

A: PROOF :

Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: Show that if a, b are real numbers then: |a| + |b| > |a + b|

A: If a,b are real numbers, then  PT a+b≥a+b Proof: Let a,b∈ℝ Let a and b denote the absolute value of…

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 if a is an odd integer, then 4 is a factor of a^2 -1

A:

Q: Prove: if a and b are integers such that a is a factor of b,  then for any integer c, a is a factor…

A:

Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: Prove that if n is odd, then 3 is not a square in GF(5n).  In other words, prove that there is no…

A:

Q: x]+n x+n m m

A: By using definition of greatest integer function, For x∈ℝ we have, x=x+x, 0≤x&lt;1 where, x is…

Q: If a, b, and c are even integers and 3a2+9b3=c, which of these is the largest number which must be a…

A:

Q: Prove: The difference of the reciprocals of two successive integers equals the product of their…

A: Given, 1n-1n+1    We have to prove the difference of the reciprocals of two successive integers…

Q: a. If p E P is a prime number, prove that the cube root of 2p is irrational.

A: We will use the basic knowledge of number theory and logic to answer this question. Note that only…

Q: Prove the statement below: The square of any integer has the form 4k or 4k + 1 for some integer k.

A:

Q: Problem 2. Prove that if integers a and b are divisible by an integer c, c+0, then a + is divisible…

A:

Q: Prove the given statement: The sum of an integer and its square is even.

A:

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

A: See the detailed solution below.

Q: Show directly that for p a prime integer, p can be written as p=(a)^2 - d(b)^2 then d is a perfect…

A: From the given statement, p be the prime factor and p can be written as p = a2 – db2.To show: d is a…

Q: Prove the given statement: The sum of the squares of two odd integers cannot be a perfect square.

A:

Q: If a, b, c are integer powers of 2, prove that equation a° + 64 .3 + 64 = c5

A: Given: Let's take a,b,c are integer powers of 2. Prove that a3+b4 =c5  does not have a solution.

Q: difference

A:

Q: Let a, b, and c be nonzero real numbers. Show, without direct evaluation, that 1 1 1 1 0. a bc ac ab

A: To show the given condition

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

A:

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

A:

Q: Attached is the question I'm needing help with. Thanks!

A:

Q: Disprove this statement by giving a counterexample. For all nonzero real numbers, a, b, c, and d,…

A: Here a,b,c,d all are non zero real .So (a/b) and (c/d) are defined.  Let,  a = 1 , b = 2 , c = 3 , d…

Q: The subject is DISCRETE MATH   Give a direct proof of the theorem “The sum of an even integer and…

A: Assume even and odd integers p and q respectively. Thus, p and q can be written as: p=2m and q=2n+1…

Q: Prove that the square of any odd integer n, (n =4q +3 for some integer q) has the form 8m + 1 for…

A: Here we have to prove that any odd integer n which is of the form (n = 4q + 3) , where q is an…

Q: 2 There are integers m and n such that m² +n² is a perfect square.

A:

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…