6. Let p be a prime number and k an integer such that x 2 + kx + p = 0 has two positive integer solutions. What is the value of k+ p? 1 O -1 O 2 O -2

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

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: If n is an even postive integer, then 3n+2^n-2 is odd.

A: Number problem....

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

A:

Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

A:

Q: 40. Ifn is any positive integer, then (n- 1)(n-2Xn - 3) ... (3(2X1) = A e B. (n-1y C. n! D. (n-1

A:

Q: , 7/4 /213, 33 b) Solve e In(9) + 21 = 41 for æ. a) Write In(- -) in the form a lIn(7) + bln(3)…

A:

Q: b) Prove that 20 15 k Σ 3.2-Σ 96.2 %D i=5 k=0

A:

Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: 6. Compute 1,= de for p= 0,1 10

A:

Q: Which statement is not correct? Sum of two irrational numbers is irrational. O For every integer n,…

A: The incorrect statement is : sum of two irrational numbers is irrational.   the detailed solution is…

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: (a) Find two integers satisfying the equation r³ + 24x² – 2 = 0 mod 23.

A:

Q: 23 - where a and b are positive integers; How many b values are there for the condition 2a + 3b =…

A:

Q: 7. For all odd integers x and y, x² +3y² = 0 (mod8) . 8. The equation x* +x² +1= 0 has no real…

A:

Q: contradiction

A: Please refer to the image below

Q: 3. (a) How many solutions are there of the equation x + y + z + t = 14, where x,y,z,t are…

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: 1) a.The set of all whole numbers not greater than 5. b.The set of all counting numbers less than or…

A: To write give set into listing method

Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

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: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: 2. Possible whenn A- Zero b. Negutive integer C- Negative Yational d- Positive integer

A: We know that Laplace transform of

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: 2. How many positive integer solutions are there to Xị + x2 + x3+ x4 < 100

A:

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: м/с How many solutions are there to the equation X1+X2 + X3 + X 4 + X5 = 25 where each Xj is and…

A:

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: Solve for n if nEN: (4+8+12+ ... +4(n-1)) + (2+4.5+7+. .. 4+5n -%3D126.5 2

A: Sum of n natural numbers is n(n+1)/2.

Q: Suppose ỹ = cje (:)- + czetx () is the general solution to = Aj. What is A O We don't have enough…

A: Use given differential equation y' = A y to find it.

Q: 3. The v77 is between what two integers?

A:

Q: 7. For all odd integers x and y, x² +3y = 0 (mod8) . 8. The equation x* +x² +1=0 has no real…

A: # 7   for all odd integer x &amp; y          x2+3y2≡0 mod 8       It is not true for all odd…

Q: 5.46. Prove that there exists a unique real number solution to the equation x O between a 2/3 and r…

A:

Q: 27=31 mod 5     True or false and how you work the problem

A: The value of a mod b is equal to the remainder from the division of a by b. Given: 27 = 31 mod 5.

Q: how many integer solutions are there to the equation  X + y + z = 10 for which  a. x,y and z + 10…

A: The positive integers are all the numbers that are greater than 0. The negative integers are the…

Q: 7. How many integer solutions are there to 2r, + 2tz + 2rz + Xg +X5 = 9 with x; 2 0? (Hint: break…

A: Solutions to the equation are the set of values for which equations returns the same value on both…

Q: 17. Given the set {x| – 2 <.x < 2, where x is an integer}, what is the solution of -2(x – 5) < 10?…

A:

Q: 7. How many non-negative integer solutions are there to the equation X1 + X2 + X3 + 4 = 16 such that…

A:

Q: 9. One merchant has seven Asava horses (A), a second has nine Haya horses (H), and a third has ten…

A: Given, One Merchant has seven Asava horses (A), a second has nine Haya horses (H) and third has ten…

Q: 11. How many solutions are there to the equation x₁ + X₂ + X3 + X₁ + X5 + x6 = 29, nonnegative…

A: We have given equation: x1+x2+x3+x4+x5+x6=29

Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

A:

Q: positive

A: Given:       integers ±1 , ±2 , ±3, ±4Sum of four positive integer = + 10Sum of four Negative…

Q: Which of the following is the solution set of In(5x – 7) < In(2x + 8) (5, 00) (-00, 5] (-00, 5) [5,…

A:

Q: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

A: Suppose that a and b are integers that divide the integer 'i'If a and b are relatively prime. Show…

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

A:

Q: How many non-negative integer solutions are there for the following equation such that 0 < x1 < 10?…

A:

Q: 5. Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  100≤n≤999 such that the last three digits of n2 is same as n.