Without using a calculator, perform the following computations (mod 7). Show all your work, and write your answer as an integer between 0 and 6. (a) 2 (36-705) (mod 7) (b) 7000000001 + (70000000027000000068) (mod 7) (c) 3² (mod 7)
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- Using Pyhton, do you think that the problems below are also solvable without using iterations? Why or why not? Printing integers from n to 1 Computing for exponents Computing for the square root Guessing a randomly-generated integerBuild truth tables or abbreviated truth tables (on your own scratch paper) to determine whether the following arguments are valid or invalid. (A v B) --> (C • D), (X v ~Y) --> (~C • ~W), (X v Z) --> (A • E), ∴∴~X Group of answer choices A. Valid B. InvalidThe following equations summarize the Euclidean algorithm with 55 and 40 as inputs. Use this information to express the gcd of 55 and 40 in the form 55a + 40b where a and b are integers. Show your work. 55 = 1·40+15, 40 = 2·15+10, 15 = 1·10+5, 10 = 2·5+0.
- Explain how to multiply two complex numbers, a + ib and c+ id, using just three multiplications of real numbers, where a, b, c, and d are real numbers and I (= v-1) is an imaginary number. The algorithm must accept the inputs a, b, c, and d and create the real component ac – bd and the imaginary component ad + bc separately.Input: (1) A real number x0, (2) n integer coefficients {a0, a1, …, an-1} Output: P(x0)=a0+a1x0+a2x02+…+an-1x0n-1 (a) Give or cite an efficient EREW PRAM algorithm (idea) to output P(x0). (b) What is the total time needed in your algorithm? Please explain the time needed for each part.As calculating floating-point arithmetic is done by a digital computers,there is often some mistake, as in all complex numbers have an inherent irrational component.explain?
- Is the set of all real numbers whose decimal expansions are computed by a machine countable? Give a justification as to why.Recall the Babylonian Algorithm for calculating a square root that we discussed in class. What will output if we were solving the square root of 21 (S = 21) using the Babylonian Algorithm with a first guess of x = 10 with just one iteration? What will the algorithm output be after a second iteration? (Answer rounded to 2 decimal places)Perform the following operations and write the final answer for: (a + b)0 (a + b)1 (a + b)2 (a + b)3 (a + b)4 (a + b)5 a) What do you notice about the coefficients? b) What do you notice about the exponents of a and b c) Could determine WITHOUT detailed (step-by-step) operations what (a + b)6 would be equal to?
- There are four basic phases of the algorithm for addition and subtraction: First, we need to check for zeroes Align the significands Add or subtract the significands Then, normalize the result Show how the following floating-point additions are performed (where significands are truncated to 4 decimal digits). Explain the solution by using the phases above. (a)5.566×10^(2 )+7.777×10^2 (b)3.344×10^1+ 8.877×10^(-2)Consider a base 26 number system wherein the letters of the alphabet are the digits. That is, A=0, B=1, C=2, … Z=25 in base 10. Use the MAL as a number in the base 26 system, and KHA as another number in the base 26 system.Add these numbers together to obtain the sum in based 26. Example 1 — if your first name is “Peter” and your surname is “Pan”, then add up PET26 and PAN26, and show the sum in base 26.Example 2 — if your first name is “Peter” and your surname is “Pa”, then add up PET26 and PAA26, and show the sum in base 26.List the first six terms of the following sequences: a. the sequence whose nth term is n! − 10 for the first n positive integers b. the sequence whose nth term is 2n − n-1 for the first n positive integers, answers must be exact rational numbers, NOT decimal approximations c. the sequence whose nth term is (-3)n + 2n for the first n positive integers d. the sequence whose nth term is ⌊nn/2⌋ for the first n positive integers, only consider positive roots e. the sequence whose first two terms are 1 and 2 and each succeeding term is the product of the two previous terms