Prove that if the integers 1, 2, 3, . . . , 65 are arranged in any order, then it is possible to look either left to right or right to left through the list and find nine numbers that are in increasing or
Homework Answers
There are many ways in which the nine digits (not including zero) can be arranged in a 3-by-3 square formation that represents a sum. For example, look at figure 1. There are also many ways to place t...
There are many ways in which the nine digits (not including
zero) can be arranged in a 3-by-3 square formation that represents
a sum. For example, look at figure 1. There are also many ways to
place the digits in a 3-by-3 grid so that, in ascending order, they
form a “rookwise” chain. In other words, moving from cell to cell
without doubling back and without moving diagonally, we can trace
through the numbers in order for example, figure 2....
1. (a) Choose 150 integers from this list {1, 2, ..., 298}, prove that there are...
1. (a) Choose 150 integers from this list {1, 2, ..., 298}, prove that there are two integers ni, n2 such that ni|n2 or n2|n1. (b) Let n1, 12, ... , 1201 be integers. Prove there exist three in- tegers ni, nj, nk E {n1, N2, ... , n201} such that 100 can divide the differences between any two of them.
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
Problem 13. (1 point) [3 Marks] Prove the following: Show that for any given 42 integers...
Problem 13. (1 point) [3 Marks] Prove the following: Show that for any given 42 integers there exist two of them whose sum, or else whose difference, is divisible by 80.
Let S be the set of distinct ordered triples comprised of the numbers 1, 2, 3, 4. To say that the...
Let S be the set of distinct ordered triples comprised of the numbers 1, 2, 3, 4. To say that the triple is distinct means that no number occurs twice in the triple. To say that the triple is ordered means that two triples in which the same numbers appear in a different order are considered to be different triples. Some of the elements of S are: 1,2,3), (1,2,4), (3,2,1), (3,2,4), (4,2,1), (4,3,2) We wish to list all of the...
Using Python3 Write a function that takes an array of integers and returns it in increasing...
Using Python3 Write a function that takes an array of integers and returns it in increasing order. There can be negative numbers, but they are treated as positive numbers. Example list= list=[-6,7,6,7,-9,1,0,-3,-2,1,4] Answer list= [0, 1, 1, -2, -3, 4, 6, -6, 7, 7, 3]
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Prove by Induction 24.) Prove that for all natural numbers n 2 5, (n+1)! 2n+3 b.)...
Prove by Induction
24.) Prove that for all natural numbers n 2 5, (n+1)! 2n+3 b.) Prove that for all integers n (Hint: First prove the following lemma: If n E Z, n2 6 then then proceed with your proof.
Prove by induction that the sum of any sequence of 3 positive consecutive integers is divisible...
Prove by induction that the sum of any sequence of 3 positive consecutive integers is divisible by 3. Hint, express a sequence of 3 integers as n+(n+1)+(n+2).
n sensors all having range equal to 2, form a unit line graph arranged on a...
n sensors all having range equal to 2, form a unit line graph arranged on a line such that the ith sensor has x-coordinate equal to xi = i - 1/i, for i = 1; 2; ....... n. 1. [1 pt] List by coordinate the first 5 vertices (left to right) of the resulting graph. 2. [1 pt] List by coordinate pairs the first 5 edges (left to right) of the resulting graph. 3. [1 pts] Is the resulting graph...
There are many ways in which the nine digits (not including
zero) can be arranged in a 3-by-3 square formation that represents
a sum. For example, look at figure 1. There are also many ways to
place the digits in a 3-by-3 grid so that, in ascending order, they
form a “rookwise” chain. In other words, moving from cell to cell
without doubling back and without moving diagonally, we can trace
through the numbers in order for example, figure 2....
1. (a) Choose 150 integers from this list {1, 2, ..., 298}, prove that there are two integers ni, n2 such that ni|n2 or n2|n1. (b) Let n1, 12, ... , 1201 be integers. Prove there exist three in- tegers ni, nj, nk E {n1, N2, ... , n201} such that 100 can divide the differences between any two of them.
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
Problem 13. (1 point) [3 Marks] Prove the following: Show that for any given 42 integers there exist two of them whose sum, or else whose difference, is divisible by 80.
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Prove by Induction
24.) Prove that for all natural numbers n 2 5, (n+1)! 2n+3 b.) Prove that for all integers n (Hint: First prove the following lemma: If n E Z, n2 6 then then proceed with your proof.
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