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 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). a2 az a, "where X Pk If a number n = p,' P2' P3 Pk-1' P¿are prime a d2 dz .. -1' a,are the corresponding exponents of the prime numbers, how x P2 X P3 k-1 numbers and 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: 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: 2. U={x|x is an integer, 2<x<10}. A and B are subsets of U such that A= {multiples of 3}, B={factors…

A: Using set theory we have to answer following.

Q: 16. Given an odd integer a, establish that a² + (a + 2)²+(a + 4)² + 1 is divisible by 12.

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Q: * 63. Let a¡, az» ·… , A, be positive real numbers. The arith- metic mean of these numbers is…

A: The arithmetic mean AM=a1+a2+...+annThe geometric mean GM=(a1,a2...,am)1n We need to show A≥G.

Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

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Q: , Let a be a positive integer, where the positive difference of every two digits of a in base 10 are…

A: Here, For 1113,     | 1-1 | =0 , | 1-3 | = |-2| = 2 For 2000,     |2-0|= 2, |0-0|= 0 For 2022,…

Q: Determine, with justification, whether the series converges absolutely, converges conditionally, or…

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Q: 23. If 6p-4 is divisible by 7, where p is an integer, which of the following is also divisible by 7.…

A: Option a is correct answer

Q: 10. Every positive integer greater than 1 has at least two divisors and can be written as a unique…

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Q: Let c, ax, be nonzero real numbers such that e. Find a given that ac² n=3 16 Ch Türkçe: c, a, 3…

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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: Let c, a, B be nonzero real numbers such that c + B. Find a given that 122n-3 2 cn-4 ac? n-D3 с — В…

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Q: b. Determine whether the statement is true or false and justify your answer. Determine which of the…

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Q: 3. Let x be an integer. Use the previous result to prove the following corollaries. (a) If x divides…

A: Divisibility: If a and b are integers, we say that a divides b, and write a|b, if there is an…

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: 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 the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: 10. Let m and n are two positive integers with m > n. Let a be a positive integer. Prove that a2" +1…

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Q: 6. Let x and y be integer. Prove the following x + y is odd if and only if x² + y² is odd.

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Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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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: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

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

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Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: 10n+1 + 10" +1

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Q: How many solutions are there to the equation X₁ + x₂ + x3 = 21, if each x; is a positive integer?…

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Q: Let c, a, ß be nonzero real numbers such that c + B. Find a given that 00 162n-3 ac² cn 4

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Q: 4. Does 3 divide (3k + 1)(3k + 2)(3k + 3)?

A: We will show by induction, put the value of k = 0 we find, (3x0+1)(3x0+2)(3x0+3) = 6  Which is…

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: 10. Prove that n³-n is divisible by 6 for every integer 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 a and b are positive integers, which of the following can never be the gcd(56a, 64b)? (а) 56а (b)…

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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: -. 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.