2.4.7: Show that every negative integer can be written in the form 2a +3b for some...
Use strong induction to show that every positive integer can be written as a sum of distinct powers of two (i.e., 20 = 1; 21 = 2; 22 =4; 23 = 8; 24 = 16; :). For example: 19 = 16 + 2 + 1 = 2^4 + 2^1 + 2^0 Hint: For the inductive step, separately consider the case where k +1 is even and where it is odd. When it is even, note that (k + 1)=2 is...
QlaJp Is Taise for all res a rec 5. Prove that if a, b, and c are nonzero integers such that alb, bic, and cla, then at least two of a, b, and c are equal. 6. Show that there exist no nonzero real numbers a and b such that Vak + ba 7. Since 2. 1+3.1+5.1 (a) Prove that there do not exist three positive integers a, b, and c such that 2a+36+ Se 11. (b) Use Mathematical Induction...
In the chemical equation 2A+3B⟶C+4D, how many molecules of C can form if 6molecules of A react?
Prove that there exists infinitely many numbers of the form an = n(n+1)/2 , for some positive integer n, such that every pair an, am (for n != m) are relatively prime. [Hint: Assume there exists a finite sequence an1 < an2 < an3 < . . . < anm, where nj are increasing positive integers. Show that using those numbers we can construct a new number that fulfills the requirements.]
Letf: AB be a function and A1.A2 CAbe subsets of the domain. Show that fAinA2) fAANAA2) a. b. Can you find a condition on fx so that in this formula could be replaced byExplain. c. If m,n are integers and n is positive, prove the following identitty: d. Show that log(n!)-O(nlogn) e. An integerm e Z is called a composite number if m is divisible by some other integere d1. For an integer numbers 2 2, show that all of...
Show that every positive integer n, there is a string of n consecutive integers where first integer is even, the second is divisible by a perfect square(other than 1), the third by a perfect cube(other than 1), etc..., and the nth is divisible by the nth power of an integer(other than 1). Then find an example for n = 3.
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
5. Find the parametric form of the solutions of the system 2a +3b+c-1 a+b+c3 Ba+4b+2c
please answer questions #7-13 7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then