Given that x and y are integers such that 0 < x < y < 9 and that the integer 77265x597y is divisible by 12, find x and y.
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Given that x and y are integers such that 0 < x < y < 9 and that the integer 77265x597y is divisible by 12, find x and y.
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- Let x and y be integers such that x = 3 (mod 10) and y = 5 (mod 10). Find the integer z such that 97x + 3y³ z (mod 10) and 0 ≤ z ≤9.Prove that there are real numbers a and b such that √a + b = √a + √b.Prove that if n is a prime number, then √n is an irrational number. (The following theorem would be useful in your proof: Theorem: Let q, r be integers and p a prime number. If p | qr, then p | qor pr.)
- You have to run Prim's algorithm for the problem defined by adjacency matrix: 1 2 3 4 5 6 7 8 9 1 0 10 9 999 999 17 999 999 999 2 3 69 10 0 14 4 2 999 999 13 999 14 0 7 999 999 999 999 999 4 999 4 7 0 999 2 8 999 999 5 999 2 999 999 0 6 999 1 999 6 17 999 999 2 6 0 999 7 999 7 999 999 999 8 999 999 0 11 4 8 999 13 999 999 1 7 11 9 999 999 999 999 999 999 4 80 8 0 1. We started from the vertex vl, so initially we have Y = {v1}: initial 1 2 3 4 5 6 7 8 9 nearest 1 1 1 1 1 1 1 1 1 distance -1 10 9 999 999 17 999 999 999 Print out the values stored in the nearest and distance arrays after first iteration of Prim's algorithm. Specify the value of vnear and the next vertex that has to be added to Y Hint: use (copy) the table above to record your answer.Let A, B and C are matrices of dimension 50 x 10, 10 x 30 and 30 x 20, respectively. What is the possible minimum number of scalar multiplications for ABC? Lütfen birini seçin: O A. 45000 O B. 30000 O C. 1500 O D. 16000 O E. 15000Prove by mathematical induction that the sum of the cubes of the first n positive integers is equal to the square of the sum of these integers.
- 1. Prove that the product of two odd integers is odd.Suppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1Let A be the set of all strings of decimal digits of length 5. For example 24157 and05189 are strings in A.(a) How many strings in A have exactly one 7?(b) How many strings in A have at least two 2’s?(c) How many strings in A have the digits in a strictly decreasing order? For example97643 and 54321 are such strings, but 14820 and 95421 are not.
- Consider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?6. Prove that if m and n are integers and mn is even, then m is even or n is even.Suppose that a Professor were to develop a method of multiplying two 12 x 12 matrices using 150 scalar multiplications and a constant number of scalar additions and subtractions: What is the recurrence equation that describes the resulting divide and conquer algorithm for multiplying two n x n matrices? And what is the asymptotic solution of the equation (use big O notation)?