2. Give an example to show that the following conjecture is not true: Every positive integer can be written in the form p + a², where p is either a prime or 1, and a > 0.

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

A:

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

A:

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: Prove that the statement is true for every positive integer n.

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

A:

Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: what is (a'+b,a+b) Wher e a and b are relatively prime integers that are not both Zero1

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: (2) Show that for any integers a,b we always have a? – 4b # 2. -

A:

Q: Prove that the statement is true for every positive integer n.

A: To prove: 1+4+7+...+3n-2=n3n-12

Q: ! Prove the following statements. (a) If k is a positive integer, then 3k+2 and k+1 are relatively…

A:

Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To prove: For every positive integer, 43+44+......+4n=4(4n-16)3

Q: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

A: Proving a statement by contradiction means assume the negation of statement true and prove the…

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

A:

Q: Prove the following statement by contradiction: If (n – 3)2 is an odd integer, then n is an even…

A:

Q: Prove that the statement is true for every positive integer n.

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n k +1 j=k

A: Recall that KCk+1=0. We will use it in the proof.

Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

A:

Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To side equations are given, 4 + 42 + 43 + · · · + 4n=4(4n-1)/3We need to find out if above equation…

Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: Prove that the statement is true for every positive integer n.

A: we have to prove that the statement is true for every positive integer n.…

Q: Prove that the statement is true for every positive integer n.

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

Q: Prove that 3" > n2 for every positive integer n.

A:

Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

A:

Q: Prove that the statements here are true for every positive integer n. a + (a + d) + (a + 2d) + · · ·…

A: To Determine: Prove that the statements here are true for every positive integer n. Given: we have…

Q: Prove that if ni is a positive integer, a± 0

A:

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

A:

Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

A:

Q: Prove that 3 divides n3+2n whenever n is a positive integer

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

Q: show that no integer of the form n^3+1 is a prime other than 2

A:

Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

A:

Q: Prove that the statement is true for every positive integer n.

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

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

A:

Q: Prove the following by contrapositive: If x is an even integer, then x^2 is even.

A:

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

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: disprove the following statement. The sum of an integer and its cube such that x + x° is

A: A statement is given as the sum of an integer and its cube such that x+x3 is even. The objective is…

Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: Prove that the statement is true for every positive integer n.

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity 'n +1° k +1 k j=k

A: We have to solve given problem:

Q: 1. Ifx is an even integer, then x is even.

A:

Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

A:

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A:

Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…