10. Every positive integer greater than 1 has at least two divisors and can be written as a unique product of some prime number/s with exponents. For example, 5 = 5'has two divisors (1 and 5 itself) 6 = 2' x 3' has four divisors (1, 2, 3 and 6) 16 = 2* has five divisors (1, 2, 4, 8 and 16). a, If a number n = p," × p,' × p, x. x px P,'where p,, p, P3 Pr-1 Prare prime numbers and a,, a,, a, . a a, are the corresponding exponents of the prime numbers, how many divisors does n have ?

Q: 16 M = 7 -5 Find formulas for the entries of M", where n is a positive integer. M" =

A: We have to find the entries of matrix Mn.

Q: Suppose we play a variant of poker in which a hand consists of a selection of six cards from a…

A: We will use combination  Number of way r object can be selected from n distinct objects…

Q: rd. 17. Show that every decimal palindromic integer with an even number of digits is divisible by…

A:

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

Q: Prove or disprove the following statement: There are prime numbers a, b, c such that a^2+ b^2= c^2.

A:

Q: Maximise Z = 5x1 + 6x2 Subject to X1 + x2 0 and integer

A:

Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: 4. Suppose that a positive integer is written in decimal notation as n = akak-1 a2a1ao where 0 <a;…

A: Here, we use theory of congruences.

Q: Find all solutions for x, up to congruence 7^78 (mod 79)

A:

Q: Prove by cases For any integer n, n^2 + 5 is not divisible

A:

Q: Let k be an positive integer. Prove that k(7k^2 + 5) + 6 is divisible by 6

A: Let k be an positive integer. Prove that k(7k^2 + 5) + 6 is divisible by 6

Q: Va 占>0 !! に

A:

Q: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: 13. Express 5 ‹(²) ³ 3 (*√81) ³ as a single power with a base of 9.

A:

Q: 2 2 Let A = Compute 412 - A and (412)A. 6-2

A:

Q: In 3 - In x - In(x + 5) = 0

A:

Q: Prove that among 7 positive integers, there are always two integers such that either their sun or…

A:

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: 10. If * -2 1 A -3 5 -1 3 Then cof (A) %3D -11 3 -5 -9 6. - 10 -3 1 11 -3 5 9 -6 10 3 -2 -1 None -11…

A: We have the matrix A=-210-3250-13 Now, if we have a matrix A=a11a12a13a21a22a23a31a32a33 then the…

Q: Some college graduates employed full-time work more than 40 hours per week, and some work fewer than…

A: Given information- Population mean, μ = 40 hours Sample mean, x-bar = 36 hours Standard deviation, s…

Q: 2. When the expression (156)(205) (557) is evaluated, it ends with a string of consecutive zero's.…

A:

Q: Which of the following gives the Binet form of the nth Fibonacci number? O Fn = Fn = Fn = Fn =…

A:

Q: Exercise 6.55 Let X and Y be random variables with joint density function if x, y > 0, f (x, y) =…

A:

Q: 10.2 Find a number a that has the following three properties: (i) a = 73003 (mod 3750). (ii) 1 ≤ a ≤…

A:

Q: One of the striking properties of Fibonacci numbers is that gcd(Fm, Fn) = Fd, (*) where d = gcd (m,…

A: 1. We see,        F9 = 34    [Fibonacci series : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,. . . . . ]…

Q: 9.1.5 Random variables X and Y have joint PDF 120, y 20, z+ys1, 10 otherwise. fx.x(z, v) - What is…

A: The joint probability function is fx, yx, y=2,   x≥0, y≥0, x+y≤10, otherwise

Q: A number with 4 digits is formed from the digits 0, 3, 6, 7, 9 without repetition. How many numbers…

A: Total numbers = 5 Since the 6 and 7 are placed next to each other consider them as a single number.…

Q: hen Let here a11 = , 923 = , 912 = " A = A-¹ a31 = = 1 1 all a21 a31 8 8 -8 , a13 = a12 a22 a32 " 0…

A:

Q: If x – 1 is an even integer, which of the following numbers must be odd? I. x(x - 1) II. 3(x - 1)…

A:

Q: 6. In an efficient way verify that the number 1 is the only positive integer number that…

A:

Q: How many positive integers with 6 or less digits are odd or divisible by 5 and do not contain any…

A: We can solve this problem by considering two cases:

Q: 10. By Binomial Theorem (2a – b)* =16a* – 32a’b+24a*b? – 8ab³ +b* | a. TRUE. b. FALSE.

A: We know the binomial therom expansion

Q: 1. Prove that for all numbers of k, (5*2^(2k+1) - 1)/3 is an odd number.

A: Let the given statement is        For k = 1 we get this as    is an odd number…

Q: Prove that n-5 is odd iff n^2-1 is odd

A: Please check step 2 for the solution.

Q: How many zero divisors are in Z,, where n = 32547?

A:

Q: Let V, E, and F be the number of vertices, number of edges, and number of faces of a cube. Consider…

A: