If m and k are positive integers, then, then Ø(m^k )= *

Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

Q: Prove that Master Cheng has won precisely 21 times in a row during some consecutive days.

A: Please find the solution

Q: 1 Show that u(7) -

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Q: Show that x2 = Y mod p has

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Q: 2. prove that

A: I solved through matrix form.

Q: Use Fermat's Theorem to find a number x between 0 and 28 with x 85 congruent to 6 modulo 29. (You…

A: as given  x85≅6mod 29   ...i as we known 29 is a prime number  by fermat's little theorem if p is a…

Q: Is given in th

A: The most repeating games used by the researchers that can be played quickly. It is discovered that…

Q: Compute C7,2. (Enter an exact number.)

A: A combination is a selection of all or part of a set of objects, without follow any order in which…

Q: Suppose that an < bn < Cn for all n, an → L, and cn → L. Show that bn → L.

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Q: Show that im n /-

A: Limit of the sequence

Q: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

Q: -, first prove that )" 6.

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: Compute [5]-l© [12] in Z17.

A: The solution is 16. (Detail is given in the handwritten notes.)

Q: 2. Prove that L(cashkt)= ki 52-K2

A: To prove that  L(cosh(kt))=ss2-k2

Q: Prove Simple disprove is Bing That Characteristic af Prime or either Zero or

A: A simple ring R is a ring with unit such that the only two-sided ideals are R and (0). The…

Q: What sequence of pseudorandom numbers is generated using the pure multiplicative generator xn+1 =…

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Q: 42 Show that E a Imod n) for all integer aif and only if n is squarefree. Ep(n) is Euler Totient…

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Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

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Q: 2 if and only if x = -| -/-

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Q: 23. Find de V100- tan O coton 2 Opbon 1

A: We have the given integral as ∫dx100-9x2 On solving the given integral furthermore, we get the…

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: Let n, m, a, b be positive integers. motl Lof oontredietion u th

A: Given, n, m, a, b be positive integers. To prove that…

Q: Prove bey wnduc hiou that for n22.

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Q: All Fermat's number is clearly odd numbers

A: Fermat number.

Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

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Q: prove that integer V intgers is an

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Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1&lt;an, n&gt;N. For n=1 a1=100011! For n=2…

Q: 27. Show that if x and y are positive real numbers that have the same first n digits in their…

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Q: : a number is even  if

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Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: 1- 9/if it of Prove that yś(-)- TT

A: In the question it is asked to prove y'(0)=π for the given function y=2xcos-1(2x)-1-4x2.

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: Show that the set of a+bi Gaussian integers has the same strength as the set of natural numbers N.

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Q: The sum of any two integers of the from 4k + 1 is always: Even O prime Zero Odd

A: If a number n is not divisible by 2 then it is an odd number.  If a number n divisible by 2 then it…

Q: Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to…

A: Given, x,y∈ℤ and x≡1mod 5, y≡2mod 5

Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

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Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: 6. Show that Autoq(Q[3], 5]) is finite.

A: To show: Autℚℚ35,53 is finite.

Q: Give an example of a pair of numbers that can be used to generate modulo residues to represent the…

A:

Q: Prove that tim 51

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show that on finite m

A: To prove that on finite measure, Lpand almost everywhere convergence are independent.

Q: For n > 3, prove that the numbers ±5, ±52, · reduced residue system modulo 2". form a

A: A subset R of the integers is called reduced a residue system modulo n if (i) gcdr,n =1 for each r…

Q: How many words of lengthn≥6 in the alphabet{1,2,3}contain at least two of each digit?(Prove a closed…

A: Here we have to find the total number of possible ways of length n≥6 in the alphabet {1,2,3} contain…

Q: -show that. Z, ax Rings.

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.