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: Use the factorization A = PDP Ak= - 7 40 - 2 11 = 45 1 1 1 01 30 to compute A I -1 5 1 - 4 1 where k…

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Q: 2) How many positive integers of at most 3 digits are there such that the product of their digits is…

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Q: u will prove that 7*+1 - 1 is divisible by 6. ypothesis, since 6 evenly divides , there exists an…

A: We will fill in the blanks.

Q: b"=1 (mod n) is valid for which of the following pair (b,n) (9.7,1729) (97,1729) (-97,1729)…

A: Answer

Q: Find the identity element for the operation * on N defined by a *b=a+b+2, Va,beN al al a 240 does…

A: Let e be the identity element. Hence, it can be written that a*e=a .

Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

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Q: 6. Let a be a number written (in base 10) as ao 10⁰+ a₁ 10¹ + a₂. 10²+...+an. 10" where 0 ≤a < 10.…

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

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Q: 3. Find the following: A. 3^0 mod 5 1 B. 3^1 mod 5 3 C. 3^2 mod 5 1 D. 3^3 mod 5 4

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

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

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Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

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Q: If fN is the set of the first 100 positive integers, A is the set of odd numbers between 50 and 90,…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

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

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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: Given A= 2n + 1-2, show that for any 4, the number of ways you can have the same value of a given…

A: Given: Objective: Show that for any value of , the number of ways to have the same value for a given…

Q: Prove the following using either direct proof, indirect proof, or proof by contradiction. 1. If n…

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Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

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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.

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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.

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

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Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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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: integer of the form okts is also but not' conversely

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

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

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Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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Q: 1. Simplify 7V8-V18. O 17/2 O 7V8 - 3V2 O 7V8-V18 O 11/2

A: NOTE: Refresh your page if you can't see any equations. .

Q: If 3x = 6 mod 9, when -2 <x< 8. Then O x=(-1, 3, 5} x=(-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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

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Q: Let m = p^3 qr^3 and n = p^2 r^5, where p,q,r are distinct primes. Find each of the following. a.…

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

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Q: If 3x = 6 mod 9, when -2 < x< 8. Then O x={-1, 3, 5} x-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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Q: -. Suppose that a positive integer is written . Prove that n is divisible by 9 if and only if y 9.

A: Here we use theory of congruence to prove our theorem.